An elementary illustrated introduction to simplicial sets

Greg Friedman
Texas Christian University
June 16, 2011
2000 Mathematics Subject Classication: 18G30, 55U10
Keywords: Simplicial sets, simplicial homotopy
Abstract
This is an expository introduction to simplicial sets and simplicial homotopy the-
ory with particular focus on relating the combinatorial aspects of the theory to their
geometric/topological origins. It is intended to be accessible to students familiar with
just the fundamentals of algebraic topology.
Contents
1 Introduction 2
2 A build-up to simplicial sets 3
2.1 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Simplicial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Delta sets and Delta maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Simplicial sets and morphisms 12
4 Realization 19
5 Products 24
6 Simplicial objects in other categories 28
7 Kan complexes 30
8 Simplicial homotopy 32
8.1 Paths and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.2 Homotopies of maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.3 Relative homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1
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8
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1
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3


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m
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]


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6

J
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n

2
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10 Concluding remarks 46
1 Introduction
The following notes grew out of my own diculties in attempting to learn the basics of sim-
plicial sets and simplicial homotopy theory, and thus they are aimed at someone with roughly
the same starting knowledge I had, specically some amount of comfort with simplicial ho-
mology (e.g. that available to those of us who grew up learning homology from Munkres [14],
slightly less so for those coming of age in Hatcher [9]) and the basic fundamentals of topologi-
cal homotopy theory, including homotopy groups. Equipped with this background, I wanted
to understand a little of what simplicial sets and their generalizations to other categories
are all about, as they seem ubiquitous in the literature of certain schools of topology. To
name just a few important instances of which I am aware, simplicial objects occur in Mays
work on recognition principles for iterated loop spaces [10], Quillens approach to rational
homotopy theory (see [16, 6]), Bouseld and Kans work on completions, localization, and
limits in homotopy theory [1], Quillens abstract treatment of homotopy theory [17], and
various aspects of homological algebra, including group cohomology, Hochschild homology,
and cyclic homology (see [21]).
However, in attempting to learn the rudiments of simplicial theory, I encountered imme-
diate and discouraging diculties, which led to serious frustration on several occasions. It
was only after several dierent attempts from dierent angles that I nally began to see
the picture, and my intended goal here is to aid future students (of all ages) to ease into
the subject.
My initial diculty with the classic expository sources such as May [11] and Curtis [3] was
the extent to which the theory is presented purely combinatorially. And the combinatorial
denitions are not often pretty; they tend to consist of long strings of axiomatic conditions
(see, for example, the combinatorial denition of simplicial homotopy, Denition 8.6, below).
Despite simplicial objects originating in very topological settings, these classic expositions
often sweep this fact too far under the rug for my taste, as someone who likes to comprehend
even algebraic and combinatorial constructions as visually as possible. There is a little bit
more geometry in Moores lecture notes [13], though still not much, and these are also
more dicult to obtain (at least not without some good help from a solid Interlibrary Loan
Department). On the other hand, there is a much more modern point of view that sweeps
both topology and combinatorics away in favor of axiomatic category theory! Goerss and
Jardine [8] is an excellent modern text based upon this approach, which, ironically, helped
me tremendously to understand what the combinatorics were getting at!
So what are we getting at here? My goal, still as someone very far from an expert in either
combinatorial or axiomatic simplicial theory, is to revisit the material covered in, roughly,
the rst chapters (in some cases the rst few pages) of the texts cited above and to provide
some concrete geometric signposts. Here, for the most part, you wont nd many complete
proofs of theorems, and so these notes will not be completely self-contained. Rather, I try
2
primarily to show by example how the very basic combinatorics, including the denitions,
arise out of geometric ideas and to show the geometric ideas underlying the most elementary
proofs and properties. Think of this as an appendix or a set of footnotes to the rst chapters
of the classic expositions, or perhaps as a Chapter 0. This may not sound like much, but
during my earliest learning stages with this material, I would have been very grateful for
something of the sort. Theoretically my reader will acquire enough of the idea to go forth
and read the more thorough (and more technical) sources equipped with enough intuition
to see whats going on.
In Section 2, we lay the groundwork with a look at the more familiar topics of simplicial
sets and, their slight generalizations, Delta sets. Simplicial sets are then introduced in Section
3, followed by their geometric realizations in Section 4 and a detailed look at products of
simplicial sets in Section 5. In Section 6, we provide a brief look at how the notion of
simplicial sets is generalized to other kinds of simplicial objects based in dierent categories.
In Section 7, we introduce Kan complexes; these are the simplicial sets that lend themselves
to simplicial analogues of homotopy theory, which we study in Section 8. This section gets
a bit more technical as we head toward more serious applications and theorems in simplicial
theory, including the denition and properties of the simplicial homotopy groups
n
(X, )
in Section 9. Finally, in Section 10, we make some concluding remarks and steer the reader
toward more comprehensive expository sources.
Acknowledments. I thank Jim McClure for his useful suggestions and Efton Park for his
careful reading of and comments on the preliminary manuscript. Later useful corrections
were suggested by Henry Adams, Daniel M ullner, and Peter Landweber.
2 A build-up to simplicial sets
We begin at the beginning with the relevant geometric notions and their immediate combi-
natorial counterparts.
2.1 Simplicial complexes
Simplicial objects are, essentially, generalizations of the geometric simplicial complexes of
elementary algebraic topology (in some cases quite extreme generalizations). So lets recall
simplicial complexes, referring the absolute beginner to [14] for a complete course in the
essentials.
Recall that a simplicial complex X is made up of simplices (generalized tetrahedra) of
various dimensions, glued together along common faces (see Figure 1). The most ecient
description, containing all of the relevant information, comes from labeling the vertices (the
0-simplices) and then specifying which collections of vertices together constitute the vertices
of simplices of higher dimension. Most often one assumes that the collection of vertices is
countable so that we can label them v
0
, v
1
, v
2
, . . ., though this is not strictly necessary - we
3
Figure 1: A simplicial complex. Note that [v
0
, v
1
, v
2
] is a simplex, but [v
1
, v
2
, v
4
] is not.
could label by v
i

iI
for any indexing set I. Then if some collection of vertices v
i
0
, . . . , v
in

constitutes the vertices of a simplex, we can write that simplex as [v

i
0
, . . . , v
in
].
A nice way to organize the combinatorial information involved is to dene the skeleta
X
k
, k = 0, 1, . . ., of a simplicial complex so that X
k
is the set of all k-simplices of X. Notice
that, having labeled our vertices so that X
0
= v
i

iI
, we can think of the elements of X
k
as subsets of X
0
; a subset v
i
0
, . . . , v
i
k
X
0
is an element of X
k
precisely if [v
i
0
, . . . , v
i
k
] is
a k-dimensional simplex of X.
If X is a complex and [v
i
0
, . . . , v
i
k
] is a simplex of X, then any subset of v
i
0
, . . . , v
i
k

is a face of that simplex and thus itself a simplex of X. This observation and this type of
notation leads us directly to the notion of an abstract simplicial complex:
Denition 2.1. An abstract simplicial complex consists of a set of vertices X
0
together
with, for each integer k, a set X
k
consisting of subsets of X
0
of cardinality k + 1. These
must satisfy the condition that any (j +1)-element subset of an element of X
k
is an element
of X
j
.
So, an abstract simplicial complex has exactly the same combinatorial information as a
geometric simplicial complex (what we have lost is geometric information about how big a
simplex is, what are its dimensions, how is it embedded in euclidean space, etc.), but we have
retained all of the information necessary to reconstruct the complex up to homeomorphism.
It is straightforward that a geometric complex yields an abstract complex, but conversely, we
can obtain a geometric complex (up to homeomorphism) from an abstract one by assigning to
each element of X
0
a geometric point and to each abstract simplex [v
i
0
, . . . , v
i
k
] a geometric
k-simplex spanned by the appropriate vertices and gluing these simplices together via the
quotient topology.
Ordered simplicial complexes. A slightly more specic way to do all this is to let our
set of vertices X
0
be totally ordered, in which case we obtain an ordered simplicial complex.
When we do this, [v
i
0
, . . . , v
i
k
] may stand for a simplex if and only if v
i
j
< v
i
l
whenever
4
j < l. This poses no undue complications as each collection v
i
0
, . . . , v
i
k
of cardinality
k still corresponds to at most one simplex. Were just being picky and removing some
redundancy in how many ways we can label a given simplex of a simplicial complex.
Example 2.2. Of course the prototypical example of a simplicial complex is the n-simplex
itself, denoted [
n
[ (see Figure 2); it will become clear later why we want to employ the
notation [
n
[ instead of just
n
.
Figure 2: The standard 0-, 1-, 2-, and 3-simplices
The n-simplex is so fundamental that one often labels the vertices simply with the num-
bers 0, 1, . . . , n. This notation should be suggestive when compared with the simplices
[v
i
0
, . . . , v
in
] appearing within more general simplicial complexes, and it is worth pointing
out at this early stage that one can think of any such simplex in a complex X as the image
of [
n
[ under a simplicial map taking 0 to v
i
0
, and so on; we will discuss simplicial maps
more formally in a moment. If X is an ordered complex, then there is precisely one way to
do this for each simplex of X. Thus another point of view on ordered simplicial complexes
is that they are made up out of images of the standard simplices (Figure 3). This will turn
out to be a very useful point of view as we progress.
Figure 3: [v
2
, v
3
, v
4
] as the image of [
2
[
Note also that the faces of the standard n-simplex can be represented simply as collections
of numbers [j
1
, . . . , j
m
] with 0 j
1
< j
2
< < j
m
n and, as an ordered simplicial
complex, the set of simplices (of all dimensions) in [
n
[ is equivalent to the collection of
ordered subsets of 0, . . . , n.
Face maps. Another aspect of ordered simplicial complexes familiar to the student of
basic algebraic topology is that, given an n-simplex, we would like a handy way of referring
5
to its (n 1)-faces. This is handled by the face maps. On the standard n-simplex, we have
n + 1 face maps d
0
, . . . , d
n
, dened so that d
j
[0, . . . , n] = [0, . . . , , . . . , n], where, as usual,
the denotes a term that is being omitted. Thus applying d
j
to [0, . . . , n] picks out the
(n 1)-face missing the vertex j (see Figure 4).
Figure 4: The face maps of [
2
[
Within more general simplicial complexes, we make the obvious extension: if [v
i
0
, . . . , v
in
]
X
n
is a simplex of the complex X, then d
j
[v
i
0
, . . . , v
in
] = [v
i
0
, . . . , v
i
j
, . . . , v
in
]. Assembled
all together, we get, for each xed n, a collection of functions d
0
, . . . , d
n
: X
n
X
n1
. Note
that here is where the ordering of the vertices of the simplices becomes important.
If one wanted to be a serious stickler, we might be careful to label the face maps from
X
n
to X
n1
as d
n
0
, . . . , d
n
n
, but this is rarely done in practice, for which we should probably
be grateful. Thus d
j
is used to represent the face map leaving out the jth vertex in any
dimension where this makes sense (i.e. dimensions j).
Furthermore, one readily sees by playing with [
n
[ that there are certain relations sat-
ised by the face maps. In particular, if i < j, then d
i
d
j
= d
j1
d
i
. Indeed, d
i
d
j
[0, . . . , n] =
[0, . . . , , . . . , , . . . , n] = d
j1
d
i
[0, . . . , n] (notice the reason that we have d
j1
in the last
expression is that removing the i rst shifts the j into the j 1 slot).
Clearly, the relation d
i
d
j
= d
j1
d
i
must hold for any simplex in a complex X (which is
made up of copies of [
n
[). This relation will become one of the axioms in the denition of
a simplicial set when we get there.
Another observation that will come up later is that there are more general face maps.
We could, for example, assign to [0, 1, 2, 3, 4, 5, 6] the face [1, 3, 4], and we could dene such
general face maps systematically. However, any such face can be obtained as a combination
of face maps that lower dimension by 1. For example, we can decompose the map just
described as d
0
d
2
d
5
d
6
. It may entertain the reader to use the face map relations and some
basic reasoning to show that any generalized face map can be obtained as a composition
d
i
1
d
im
uniquely if we require that i
j
< i
j+1
for all j.
6
2.2 Simplicial maps
The heart of the transition (in my opinion) from simplicial complexes to simplicial sets is
tied up in simplicial maps.
Recall (see [14, Section 2]) that if K and L are simplicial complexes, then a simplicial
map f : K L is determined by taking the vertices v
i
to vertices f(v
i
) of L such that if
[v
i
0
, . . . , v
i
k
] is a simplex of K then f(v
i
0
), . . . , f(v
i
k
) are all vertices (not necessarily unique)
of some simplex in L. Given such a function K
0
L
0
, the rest of f : K L is determined
by linear interpolation on each simplex (if x K can be represented by x =

n
j=1
t
j
v
i
j
in
barycentric coordinates of the simplex spanned by the v
i
j
, then f(x) =

n
j=1
t
j
f(v
i
j
)). The
resulting function f : K L is continuous (see [14]).
Example 2.3. We have already seen one example of a simplicial map in Example 2.2, the one
that takes the n-simplex [
n
[ to a simplex [v
i
0
, . . . , v
in
] of some simplicial complex.
Example 2.4. The interesting feature of simplicial maps, from the point of view of simplicial
sets, is that simplicial maps can collapse simplices. For example, consider the simplicial map
f :
2

1
determined by f(0) = 0, f(1) = 1, f(2) = 1 that collapses the 2-simplex down
to the 1-simplex (see Figure 5). The great benet of the theory of simplicial sets is a way to
generalize these kinds of maps in order to preserve information so that we can still see the
image of the 2-simplex hiding in the 1-simplex as a degenerate simplex (see Section 3).
Figure 5: A collapse of [
2
[ to [
1
[
2.3 Delta sets and Delta maps
Delta sets (sometimes called -sets) constitute an intermediary between simplicial com-
plexes and simplicial sets. These allow a degree of abstraction without yet introducing the
degeneracy maps we have begun hinting at.
7
Denition 2.5. A Delta set
1
consists of a sequence of sets X
0
, X
1
, . . . and, for each n 0,
maps d
i
: X
n+1
X
n
for each i, 0 i n + 1, such that d
i
d
j
= d
j1
d
i
whenever i < j.
Of course this is just an abstraction, and generalization, of the denition of an ordered
simplicial complex, in which the X
n
are the sets of n-simplices and the d
i
are the face maps.
However, there are Delta sets that are not simplicial complexes:
Example 2.6. Consider the cone C obtained by starting with the standard 2-simplex [
2
[ =
[0, 1, 2] and gluing the edge [0, 2] to the edge [1, 2] (see Figure 6). This space is no longer a
simplicial complex (at least not with the triangulation given), since in a simplicial complex,
the faces of a given simplex must be unique. This is no longer the case here as, for example,
the edge [0,1] now has both endpoint vertices equal to each other.
Figure 6: Gluing [
2
[ into a cone
However, this is a Delta set. Without (I hope!) too much risk of confusion, we use the
notation for the simplices in the triangle to refer also to their images in the cone. So, for
example [0] and [1] now both stand for the same vertex in the cone and [0, 1] stands for the
circular base edge. Then C
0
= [0], [2], C
1
= [0, 1], [0, 2], C
2
= [0, 1, 2], and C
n
= for
all n > 2. The face maps are the obvious ones, also induced from the triangle, so that, e.g.
d
2
[0, 1, 2] = [0, 1] and d
0
[0, 1] = d
1
[0, 1] = [0] = [1]. It is not hard to see that the axiom on
the d
i
is satised - it comes right from the fact that it holds for the standard 2-simplex.
Example 2.7. One feature of Delta sets we need to be careful about is that, unlike for
simplicial complexes, a collection of vertices does not necessarily specify a unique simplex.
For example, consider the Delta set with X
0
= v
0
, v
1
, X
1
= e
0
, e
1
, d
0
(e
0
) = d
0
(e
1
) = v
0
,
and d
1
(e
0
) = d
1
(e
1
) = v
1
. Both 1-simplices have the same endpoints. See Figure 7.
Figure 7: A Delta set containing two edges with the same vertices
1
It seems to be at least fairly usual to capitalize the word Delta in this context, probably since it
is essentially a stand-in for the Greek capital letter . However, for reasons that will become clear, it is
probably best to avoid the notation -set and to use instead the English stand-in.
8
Thus Delta sets aord some greater exibility beyond ordered simplicial complexes. One
may continue to think of the sets X
n
as collections of simplices and interpret from the face
maps how these are meant to be glued together (Exercise: Give each simplex of the cone
X of the preceding example an abstract label, write out the full set of face maps in these
labels, then reverse engineer how to construct the cone from this information. One sees that
everything is forced. For example, there is one 2-simplex, two of whose faces are the same,
so they must be glued together!). However, it is common in the fancier literature not to
think of the X
n
as collections of simplices at all but simply as abstract sets with abstract
collections of face maps. At least this is what authors would have us believe - I tend to
picture simplices in my head anyway, while keeping in mind that this is more of a cognitive
aid than it is whats really going on.
The category-theoretic denition. While were walking the tightrope of abstraction,
lets take it a step further. Recall that we discussed in Example 2.2 that we can think of an
ordered simplicial complex as a collection of isomorphic images of the standard n-simplices.
Of course to describe the simplicial complex fully we need not only the images of the standard
simplices but we need to know which image faces are attached together. This information
is contained in the face maps, which tell us when two simplices share a face. Theres an
alternative denition of Delta complexes that takes more of this point of view. It might be
a little scary if youre not that comfortable with category theory, but dont worry, Ill walk
you through it (though I do assume you know the basic language of categories and functors).
First, we dene a category

:
Denition 2.8. The category

has as objects the nite ordered sets [n] = 0, 1, 2 . . . , n.
The morphisms of

are the strictly order-preserving functions [m] [n] (recall that f is
strictly order-preserving if i < j implies f(i) < f(j)).
Of course the objects of

should be thought of as our prototype ordered n-simplices. The
morphisms are only dened when m n, and you can think of these morphisms as taking an
m-simplex and embedding it as a face of an n-simplex (see gure 8). Note that, since order
matters, there are exactly as many ways to do this as there are strictly order-preserving
maps [m] [n].
Next, we think about the opposite category

op
. Recall that this means that we keep
the same objects [n] of

, but for every morphism [m] [n] in

, we instead have a map
[n] [m] in

op
. What should this mean? Well if a given morphism [m] [n] was the
inclusion of a face, then the new opposite map [n] [m] should be thought of as taking the
n-simplex [n] and prescribing a given face. This is just a generalization of what we have seen
already: if we consider in

the morphism D
i
: [n] [n + 1] dened by the strictly order-
preserving map 0, . . . , n 0, . . . , , . . . , n + 1, then in

op
this corresponds precisely to
the simplex face map d
i
. Even better, it is easy to check once again that, with this denition,
d
i
d
j
= d
j1
d
i
when i < j, simply as an evident property of strictly order-preserving maps.
This is really how we argued for this axiom in the rst place!
So, in summary, the category

op
is just the collection of elementary n-simplices together
with the face maps (satisfying the face map axiom) and the iterations of face maps. But
9
Figure 8: A partial illustration of the category

this should be precisely the prototype for all Delta sets:

Denition 2.9 (Alternative denition for Delta sets). A Delta set is a covariant functor
X :

op
Set, where Set is the category of sets and functions. Equivalently, a Delta set is
a contravariant functor

Set.
Lets see why this makes sense. A functor takes objects to objects and morphisms to
morphisms, and it obeys composition rules. So, unwinding the denition, a covariant functor

op
Set assigns to [n]

op
a set X
n
(which we can think of, and which we refer to,
as a set of simplices) and gives us, for each strictly order-preserving [m] [n] in

(or
its corresponding opposite in

op
) a generalized face map X
n
X
m
(which we think of
as assigning an m-face to each simplex in X
n
). As noted previously, these generalized face
maps are all compositions of our standard face maps d
i
, so the d
i
(and their axioms) are the
only ones we usually bother focusing on.
So what just happened? The power of this denition is really in its point of view. Instead
of thinking of a Delta set as being made up of a whole bunch of simplices one at a time, we
can now think of the standard n-simplex as standing for all of the simplices in X
n
, all at
once - the functor X assigns to [n] the collection of all of the simplices of X
n
(see Figure 9).
The face map d
i
applied to the standard simplex [n] represents all of the ith faces of all the
n-simplices simultaneously.
At the same time, we see how any argument in X really comes from what happens back
in

. The commutativity axiom d
i
d
j
= d
j1
d
i
in a Delta set X is just a consequence of this
being true in the prototype simplex [n] and inherent properties of functors. Well get a lot of
10
Figure 9: A Delta complex as the functorial image of

mileage out of this kind of thinking: things wed like to prove in a Delta set X can often be
proved just by proving them in the prototype standard simplex and applying functoriality.
Delta maps. We wont dwell overly long on Delta maps, except to observe that they, too,
point toward the need for simplicial sets (however, see [18] where Delta complexes and Delta
maps are treated in their own right).
Going directly to the category theoretic denition, given two Delta sets X, Y , thought
of as contravariant functors

Set, a morphism X Y is a natural transformation of
functors from X to Y . Thus a morphism consists of a collection of set maps X
n
Y
n
that
commute with the face maps.
Example 2.10. There is an apparent Delta map from the standard 2-simplex [0, 1, 2] to the
cone C of Example 2.6. See Figure 10.
Figure 10: The Delta map from [
2
[ to the cone
The astute reader will notice something shy here. We would hope that simplicial maps
of simplicial complexes would yield morphisms of Delta sets. However, consider the collapse
: [
2
[ = [0, 1, 2] [
1
[ = [0, 1] dened by (0) = 0 and (1) = (2) = 1 (see Figure
5). To be a Delta set morphism, the simplex [0, 1, 2] [
2
[
2
would have to be taken to
an element of [
1
[
2
. But this set is empty! There are no 2-simplices of [
1
[. Something is
amiss. We need simplicial sets.
11
3 Simplicial sets and morphisms
Simplicial sets generalize both simplicial complexes and Delta sets.
When approaching the literature, the reader should be very careful about terminology.
Originally ([5]), Delta sets were referred to as semi-simplicial complexes, and, once the
degeneracy operations we are about to discuss were discovered, the term complete semi-
simplicial complex (c.s.s. set, for short) was introduced. Over time, with Delta sets becoming
of less interest, complete semi-simplicial was abbreviated back to semi-simplicial and
eventually to simplicial, leaving us with the simplicial sets of today. Meanwhile, some
modern authors have returned to using semi-simplicial complexes to refer to what we are
calling Delta sets, on the grounds that, as we will see, the category is the prototype for
simplicial sets, not Delta sets, for which we have been using the prototype category

. This
all sounds very confusing because it is, and the reader is advised to be very careful when
reading the literature.
2
We try to be careful and use only the three terms simplicial complex, Delta set, and
simplicial set. In particular, be sure to note the dierence between simplicial complex
and simplicial set going forward.
Degenerate simplices. Recall from Example 2.4 that a simplicial map can collapse a
simplex. In that example, we had a simplicial map : [
2
[ [
1
[ dened on vertices so
that (0) = 0 and (1) = (2) = 1. Recall also that we have begun to think of simplicial
complexes and Delta sets as collections of images of standard simplices under appropriate
maps. Well, here is a map of the standard 2-simplex [
2
[. What image simplex does it give
us in [
1
[ under ? In the land of simplicial sets, the image ([
2
[) is an example of a
degenerate simplex.
Roughly speaking, degenerate simplices are simplices that dont have the correct
number of dimensions. A degenerate 3-simplex might be realized geometrically as a 2-
dimensional, 1-dimensional, or 0-dimensional object. Geometrically, degenerate simplices
are hidden; thus the clearest approach to dealing with them lies in the combinatorial
notation we have been developing all along.
The key both to the idea and to the notation is in allowing vertices to repeat. The natural
way to label ([
2
[) = ([0, 1, 2]) in our example is as [0, 1, 1], reecting where the vertices
of [
2
[ go under the map. This violates our earlier principle that simplices in complexes
should be written [v
0
, . . . , v
n
] with the v
i
distinct vertices written in order, but sometimes
in mathematics we need a new, more general principle. For degenerate simplices, well keep
the orderings but dispense with the uniqueness. Thus, ocially, a degenerate simplex is a
[v
i
0
, . . . , v
in
] for which the v
i
j
are not all distinct.
Example 3.1. How many 1-simplices, including degenerate ones, are lurking within the ele-
mentary 2-simplex [0, 1, 2]? A 1-simplex is still written [a, b], with a b, but now repetition
is allowed. The answer is six: [0, 1], [0, 2], [1, 2], [0, 0], [1, 1], and [2, 2]. See the middle picture
in Figure 11.
2
I thank Jim McClure for explaining to me this historical progression.
12
Similarly, within [
2
[ = [0, 1, 2] there are now three kinds of 2-simplices. We have the
nondegenerate [0, 1, 2], the 2-simplices that degenerate to 1-dimension such as [0, 1, 1] and
[0, 0, 2], and we have the 2-simplices that degenerate to 0-dimensions such as [0, 0, 0] and
[2, 2, 2].
Working with degenerate simplices makes drawing diagrams much more dicult. We
take a crack at it in Figure 11.
Figure 11: The rst picture represents all of the 1-simplices in [
1
[, including the degenerate
ones that are taken to individual vertices. The second picture represents all the 1-simplices
in [
2
[, and the last picture represents all of the degenerate 2-simplices in [
2
[.
As implied by the diagram, we can think of degenerate simplices as being the images of
collapsing maps such as that in Example 2.4.
Of course any simplicial complex or Delta set can be expanded conceptually to include
degenerate simplices. In the example of Figure 1, we might have the degenerate 5-simplex
[v
2
, v
2
, v
2
, v
3
, v
3
].
Notice also that our innocent little n-dimensional simplicial complexes suddenly contain
degenerate simplices of arbitrarily large dimension. Even the 0-simplex [
0
[ = [0] becomes
host to degenerate simplices such as [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0].
The situation has degenerated indeed! To keep track of it all, we need degeneracy maps.
Degeneracy maps. Degeneracy maps are, in some sense, the conceptual converse of face
maps. Recall that the face map d
j
takes an n-simplex and give us back its jth (n 1)-face.
On the other hand, the jth degeneracy map s
j
takes an n-simplex and gives us back the jth
degenerate (n + 1)-simplex living inside it.
13
As usual, we illustrate with the standard n-simplex, which will be a model for what
happens in all simplicial sets. Given the standard n-simplex [
n
[ = [0, . . . , n], there are
n + 1 degeneracy maps s
0
, . . . , s
n
, dened by s
j
[0, . . . , n] = [0, . . . , j, j, . . . , n]. In other
words, s
j
[0, . . . , n] gives us the unique degenerate n + 1 simplex in [
n
[ with only the jth
vertex repeated.
Again, the geometric concept is that s
j
[
n
[ can be thought of as the process of collapsing

n+1
down into
n
by the simplicial map
j
dened by
j
(i) = i for i < j,
j
(j) =
j
(j+1) =
j and
j
(i) = i 1 for i > j + 1.
This idea extends naturally to simplicial complexes, to Delta sets, and to simplices that
are already degenerate. If we have a, possibly degenerate, n-simplex [v
i
0
, . . . , v
in
] with i
k

i
k+1
for each k, 0 k n, then we set s
j
[v
i
0
, . . . , v
in
] = [v
i
0
, . . . , v
i
j
, v
i
j
, . . . , v
in
], i.e. repeat
v
i
j
. This is a degenerate simplex in [v
i
0
, . . . , v
in
].
It is not hard to see that any degenerate simplex can be obtained from an ordinary
simplex by repeated application of degeneracy maps. Thus, just as any face of a simplex can
be obtained by using compositions of the d
i
, any degenerate simplex can be obtained from
compositions of the s
i
.
Also, as for the d
i
, there are certain natural relations that the degeneracy maps possess.
In particular, if i j, then s
i
s
j
[0, . . . , n] = [0, . . . , i, i, . . . , j, j, . . . , n] = s
j+1
s
i
[0, . . . , n]. Note
that we have j + 1 in the last formula, not s
j
, since the application of s
i
pushes j one slot
to the right.
Furthermore, there are relations amongst the face and degeneracy operators. These are
a little more awkward to write down since there are three possibilities:
d
i
s
j
= s
j1
d
i
if i < j,
d
j
s
j
= d
j+1
s
j
= id,
d
i
s
j
= s
j
d
i1
if i > j + 1.
These can all be seen rather directly. For example, applying either side of the rst formula
to [0, . . . , n] yields [0, . . . , , . . . , j, j, . . . , n]. Note also that the middle formula takes care of
both i = j and i = j + 1.
Simplicial sets. We are nally ready for the denition of simplicial sets:
Denition 3.2. A simplicial set consists of a sequence of sets X
0
, X
1
, . . . and, for each
n 0, functions d
i
: X
n
X
n1
and s
i
: X
n
X
n+1
for each i with 0 i n such that
d
i
d
j
= d
j1
d
i
if i < j,
d
i
s
j
= s
j1
d
i
if i < j,
d
j
s
j
= d
j+1
s
j
= id, (1)
d
i
s
j
= s
j
d
i1
if i > j + 1,
s
i
s
j
= s
j+1
s
i
if i j.
Example 3.3. Every simplicial complex or Delta set can be made into a simplicial set by
adjoining all possible degenerate simplices. Conversely, each simplicial set is also a Delta set
14
by neglect of structure. However, a simplicial set does not necessarily come from a simplicial
complex as, for example, not every Delta set is a simplicial complex.
Example 3.4. The standard 0-simplex X = [0], now thought of as a simplicial set, is the
unique simplicial set with one element in each X
n
, n 0. The element in dimension n is
n+1 times

[0, . . . , 0].
Example 3.5. As a simplicial set, the standard 1-simplex X = [0, 1] already has n+2 elements
in each X
n
. For example, X
2
= [0, 0, 0], [0, 0, 1], [0, 1, 1], [1, 1, 1].
Remark 3.6. We will use
n
or [0, . . . , n] to refer to the standard n-simplex, thought of as
a simplicial set.
Example 3.7. Now for an example familiar from algebraic topology. Given a topological
space X, let S(X)
n
be the set of continuous functions from [
n
[ to X. Together with face
and degeneracy maps that we will describe in a moment, these constitute a simplicial set
called the singular set of X. The singular chain complex S

(X) from algebraic topology has

each S
n
(X) equal to the free abelian group generated by S(X)
n
.
To dene the face and degeneracy maps, let : [
n
[ X be a continuous map repre-
senting a singular simplex (Figure 12). The singular simplex d
i
is dened as the restriction
of to the ith face of [
n
[. Equivalently it is the composition of and the simplicial in-
clusion map [0, . . . , n 1] [0, . . . , , . . . , n] (Figure 13). These are precisely the same as
the terms that show up in the boundary map of the singular chain chain complex where
=

n
i=0
(1)
i
d
i
.
Figure 12: A singular simplex
On the other hand, the degeneracy s
i
takes the singular simplex to the composition
of : [
n
[ = [0, . . . , n] X with the geometric collapse represented by the degeneracy
[0, . . . , n + 1] [0, . . . , i, i, . . . , n]. Once again, a degenerate simplex is a collapsed version
of an ordinary simplex (Figure 14).
S(X) is clearly a simplicial set: a face map takes a singular simplex to the map dened
along a face of the model simplex and a degeneracy is represented by a collapsed simplex.
It is our usual model, just redesigned within the context of the continuous map .
15
Figure 13: A face of a singular simplex
Figure 14: A degenerate singular simplex
Some more examples of simplicial sets are given below in Section 4, where we can better
study their geometric manifestations.
Nondegenerate simplices.
Denition 3.8. A simplex x X
n
is called nondegenerate if x cannot be written as s
i
y for
any y X
n1
and any i.
Every simplex in the sense of Section 2 of a simplicial complex or Delta set is a nondegen-
erate simplex of the corresponding simplicial set. If Y is a topological space, an n-simplex
of S(Y ) is nondegenerate if it cannot be written as the composition
n


k

Y , where
is a simplicial collapse with k < n and is a singular k-simplex.
Note that it is possible for a nondegenerate simplex to have a degenerate face (see Exam-
ple 4.7, below, though it might be good practice to try to come up with your own example
rst). It is also possible for a degenerate simplex to have a nondegenerate face (for example,
we know d
j
s
j
x = x for any x, degenerate or not).
16
The categorical denition. As for Delta sets, the basic properties of simplicial sets
derive from those of the standard n-simplex. In fact, that is where the prototypes of both
the face and degeneracy maps live and where we rst developed the axioms relating them.
Thus it is not surprising (at this point) that there is a categorical denition of simplicial
sets, analogous to the one for Delta sets, in which each simplicial set is the functorial image
of a category, , built from the standard simplices.
Denition 3.9. The category has as objects the nite ordered sets [n] = 0, 1, 2 . . . , n.
The morphisms of are order-preserving functions [m] [n].
Notice that the only dierence between the denitions of

and is that the morphisms
in only need to be order-preserving and not strictly order-preserving. Thus, equating the
objects [n] with the simplices
n
, the morphisms no longer need to represent only inclusions
of simplices but may represent degeneracies as well. In more familiar notation, a typical
morphism, say, f : [5] [3] might be described by f[0, 1, 2, 3, 4, 5] = [0, 0, 2, 2, 2, 3], which
can be thought of as a simplicial complex map taking the 5-simplex degenerately to the
2-face of the 3-simplex spanned by 0, 2, and 3.
As in

, the morphisms in are generated by certain maps between neighboring car-
dinalities D
i
: [n] [n + 1] and S
i
: [n + 1] [n], 0 i n. The D
i
are just as for

:
D
i
[0, . . . , n] = [0, . . . , , . . . , n +1]. The new maps, which couldnt exist in

, are dened by
S
i
[0, . . . , n + 1] = [0, . . . , i, i, . . . , n]. It is an easy exercise to verify that all morphisms in
are compositions of the D
i
and S
i
and that these satisfy axioms analogous to those in the
denition of simplicial set. Later on, we will also use D
i
and S
i
to stand for the geometric
maps they induce on the standard geometric simplices.
To get to our categorical denition of simplicial sets, we must, as for Delta sets, consider

op
. The maps D
i
become their opposites, denoted d
i
, and these correspond to the face
maps as before: the opposite of the inclusion D
i
: [n] [n + 1] of the ith face is the ith
face map, d
i
, which assigns to the n-simplex its ith face. The opposites of the S
i
become
the degeneracies; the opposite of the collapse S
i
: [n + 1] [n] that pinches together the i
and i + 1 vertices of an n + 1 simplex is the ith degeneracy map, s
i
, which assigns to the
n-simplex
n
the degenerate n+1-simplex within
n
that repeats the ith vertex. See Figure
15.
Of course, the d
i
and s
i
can be checked to satisfy the axioms in the denition of simplicial
set given above.
Denition 3.10 (Categorical denition of simplicial set). A simplicial set is a contravariant
functor X : Set (equivalently, a covariant functor X :
op
Set).
The reader should compare this with the categorical denition of Delta sets and reassure
himself/herself that this denition is equivalent to Denition 3.2. As for Delta sets, the
power in this denition is that we can think of the standard n-simplex as standing for all
of the simplices in X
n
, all at once - the functor X assigns to [n] all of the n-simplices in
X
n
- and the standard face and degeneracy maps d
i
and s
i
pick out all of the faces and
degeneracies of X
n
by functoriality.
17
Figure 15: How to visualize D
i
, d
i
, S
i
, and s
i
. Our diculty with drawing degeneracies
extends here so that we represent the image of s
i
pictorially by the picture for S
i
. In other
words, the image of s
1
in the bottom right is the degenerate 2-simplex arising from the
collapse map S
1
.
Example 3.11. Lets re-examine the singular set S(Y ) of the topological space Y from this
point of view. The singular set S(Y ) is a functor Set that assigns to [n] the set
Hom
Top
([
n
[, Y ), the set of all continuous maps from [
n
[ to Y . It assigns to the face
and degeneracy maps of the face and degeneracy maps of Example 3.7, i.e. we have the
following correspondences:
[n] Hom
Top
([
n
[, Y ) [n] Hom
Top
([
n
[, Y )

[n 1]
d
i
?
Hom
Top
([
n1
[, Y )
d
i
?
[n + 1]
s
i
?
Hom
Top
([
n+1
[, Y ).
s
i
?
The reader should check that the denitions for the face and degeneracy maps of the singular
set dened above are consistent with the claimed functoriality. (Notice that the maps on the
right sides of these diagrams should more appropriately be labeled S(Y )(d
i
) and S(Y )(s
i
),
but we stick with common practice and use d
i
and s
i
for face and degeneracy maps wherever
we nd them.)
Simplicial morphisms. Simplicial sets themselves constitute a category S. In the cat-
egorical language, if X and Y are simplicial sets (and thus functors Set), then a
morphism f : X Y is a natural transformation of functors. Unwinding this to more
concrete language, f consists of set maps f
n
: X
n
Y
n
that commute with face operators
and with degeneracy operators.
Example 3.12. At last we have a context in which to explain properly the collapse map
: [
2
[ [
1
[ of Example 2.4. We extend to the map of simplicial sets :
2

1
that
takes
2
= [0, 1, 2] to the degenerate simplex [0, 1, 1] = s
1
([0, 1]). At the same time, it is
doing an innite number of other things. For example, it takes the vertex [0]
2
to [0]
1
,
it takes the vertices [1], [2]
2
to [1]
1
, it takes the 1-simplex [0, 1]
2
to [0, 1]
1
,
18
it takes the 1-simplex [1, 2]
2
to the degenerate 1-simplex
3
[1, 1] = s
0
[1]
1
, and it even
takes the degenerate simplex [0, 1, 1, 2, 2, 2] = s
4
s
3
s
1
[0, 1, 2]
2
to the degenerate simplex
s
4
s
3
s
1
[0, 1, 1] = [0, 1, 1, 1, 1, 1]
1
. And much much more.
Example 3.13. Notice that, unlike simplicial maps on simplicial complexes, morphisms on
simplicial sets are not completely determined by what happens on vertices. For example,
consider the simplicial morphisms from
1
to the Delta set decomposition of the circle with
two distinct vertices of Example 2.7. If we have a simplicial morphism that takes [0] to [v
0
]
and [1] to [v
1
], there are still two possibilities for where to send [0, 1].
Example 3.14. On the other hand, given a map of ordered simplicial complexes f : X Y ,
this induces a map of the associated simplicial sets obtained by adjoining all naturally
occurring degenerate simplices, i.e. all simplices of the form [v
i
0
, . . . , v
in
] with i
0
i
n
,
where all v
i
j
are vertices of some particular simplex of X (or, respectively, of Y ). In this
case, a function on vertices does determine a simplicial map.
Remark 3.15. Notice that it is always enough to dene a simplicial morphism by what it
does to nondegenerate simplices. What happens to the degenerate simplices is forced by the
denition since, e.g. f(s
i
(x)) = s
i
(f(x)). Similarly, what happens on faces is forced by what
happens on the simplices of which they are faces. Thus, altogether, simplicial morphisms
can be described by specifying what they do to a rather small collection of nondegenerate
simplices.
From here on, well abandon the distinction between simplicial map and simplicial
morphism and use the terms interchangeably as applied to simplicial sets.
4 Realization
If the idea of simplicial objects is to abstract from geometry to combinatorics, there should
be a way to reverse that process and turn simplicial sets into geometric objects. Indeed that
is the case. The denition looks a bit o-putting at rst (what concerning simplicial sets
doesnt?), but, in fact, well see that simplicial realization is a very natural thing to do.
Denition 4.1. Let X be a simplicial set. Give each set X
n
the discrete topology and let
[
n
[ be the n-simplex with its standard topology. The realization [X[ is given by
[X[ = H

n=0
X
n
[
n
[/ ,
where is the equivalence relation generated by the relations (x, D
i
(p)) (d
i
(x), p) for
x X
n+1
, p [
n
[ and the relations (x, S
i
(p)) (s
i
(x), p) for x X
n1
, p [
n
[. Here D
i
and S
i
are the face inclusions and collapses induced on the standard geometric simplices as
in our discussion above of the category .
3
Careful: [1] is a 0-simplex, so s
0
is the appropriate (indeed the only well-dened) degeneracy map.
Remember that s
0
tells us to repeat what occurs in the 0th place - it doesnt know whats in that place.
19
To see why this denition makes sense, lets think about how we would like to form a
simplicial complex out of the data of a simplicial set. From the get-go, we have been thinking
of the X
n
as collections of simplices. So this is just what X
n
[
n
[ is: a disjoint collection
of simplices, one for each element of X
n
. The next natural thing to do is to identify common
faces. This is precisely what the relation (x, D
i
(p)) (d
i
(x), p) encodes (see Figure 16): The
rst term of (x, D
i
(p)) (x, [
n+1
[) is an (n+1)-simplex of X and the second term D
i
(p) is
a point on the ith face of a geometric (n+1)-simplex. On the other hand, (d
i
(x), p) is the ith
face of x together with the same point, now in a stand-alone n-simplex. So the identication
described just takes the n-simplex corresponding to d
i
(x) in X
n
[
n
[ and glues it as the
ith face of the (n+1)-simplex assigned to x in X
n+1
[
n+1
[. Since a similar gluing is done
for any other y and j such that d
j
(y) = d
i
(x), the eect is to glue faces of simplices together.
Figure 16: In the realization, the 1-simplex representing d
0
x, pictured on the right, is glued
to the 2-simplex representing x, pictured on the left, along the appropriate face.
The next natural thing to do is suppress the degenerate simplices, since theyre encoded
within nondegenerate simplices anyway. This is what the relation (x, S
i
(p)) (s
i
(x), p)
for x X
n1
, p [
n
[ does, although more elegantly. This relation tells us that given a
degenerate n-simplex s
i
(x) and a point p in the pre-collapse n-simplex [
n
[, we should glue
p to the (n1)-simplex represented by x at the point S
i
(p) in the image of the collapse map.
That still sounds a little confusing, but the idea is straightforward: [
n
[s corresponding to
degenerate n-simplices get collapsed in the natural way into the (n 1)-simplices they are
degeneracies of. See Figure 17. We note also that there is no reason to believe that x itself is
nondegenerate. It might be, in which case the simplex corresponding to x is itself collapsed.
This provides no diculty.
Example 4.2. Recall that the 0-simplex [0], thought of a simplicial set, has one simplex
[0, . . . , 0] in each dimension 0. Thus [[0][ = H

i=0
[
i
[/ . So in dimension 0 we have a single
vertex v. In dimension 1, we have a single simplex [0, 0] = s
0
[0]. The gluing instructions tell
us to identify each (s
i
(x), p) = ([0, 0], p) ([0, 0], [
1
[) with ([0], S
0
(p)) = ([0], v). Thus the
[
1
[ in dimension 1 gets collapsed to the vertex. Similarly, since each point of the 2-simplex
([0, 0, 0], [
2
[) gets identied to a point of ([0, 0], [
1
[), and so on, we see that the whole
situation collapses down to a single vertex.
Example 4.3. Generalizing the preceding example, [[0, . . . , n][ = [
n
[ is just the standard
geometric n-simplex, justifying our earlier use of notation. We encourage the reader to
explore this example on his or her own, noting that all of the degenerate simplices wind up
tucked away within actual faces of [
n
[, just where we expect them.
20
Figure 17: In the realization, the 2-simplex representing s
1
x, pictured on the right, is glued
to the 1-simplex representing x, pictured on the left, via the appropriate collapse, depicted
by S
1
.
Example 4.4. More generally, given any simplicial complex, the realization of the simplicial
set associated to it by adjoining only degenerate simplices returns the original simplicial
complex.
Example 4.5. There is an analogous realization procedure for Delta sets. Given a Delta set
X, we can dene the realization [X[

by
[X[

= H

n=0
X
n
[
n
[/ ,
where is the equivalence relation generated by (x, D
i
(p)) (d
i
(x), p) for x X
n+1
, p
[
n
[. These realizations yield the types of spaces we have been drawing already to represent
Delta sets. These are sometimes called Delta complexes; see, e.g., [9].
However, given a simplicial set X, the simplicial set realization of X is not generally
going to be the same as the Delta set realization of the associated Delta set, say X

, that
we obtain by neglect of structure.
For example, consider the simplicial set
0
. As noted, its simplicial realization, [
0
[ is
the topological space consisting of a single point. But recall that the simplicial set
0
has
exactly one simplex in each dimension, and the neglect of structure that turns this into a
Delta set drops the degeneracy relation but still leaves a Delta set with one simplex in each
dimension and all face maps the unique possible ones. Thus the Delta set realization [
0

is an innite dimensional CW complex with one cell in each dimension whose n-dimensional
cell is attached by gluing each face of an n-simplex, in an order-preserving manner, to the
image of the unique (n 1)-simplex in the (n 1)-skeleton. Thus the 1-skeleton of [
0

is a circle, the 2-skeleton is the dunce cap (see, e.g., [2, Section 14]), and so on. This is
evidently not homeomorphic to [
0
[. (However, they are homotopy equivalent; this will be
true in general, see [18]).
In what follows, discussion of realization and the notation [X[ will refer exclusively to
simplicial set realization unless noted otherwise.
Example 4.6. Let Y be a topological space, and let S(Y ) be its singular set. [S(Y )[ will
be huge (unless Y is discrete - what will it be then?). While this looks discouraging, it turns
out that the natural map [S(Y )[ Y (which acts on the realization of each of each singular
simplex by the map dening that singular simplex) induces isomorphisms on all homotopy
21
groups; see [12, Theorem 4]. In particular, if Y is a CW complex, this is enough to assure
[S(Y )[ and Y are homotopy equivalent as a consequence of the Whitehead Theorem (see [2,
Corollary VII.11.14]), as we will see below in Theorem 4.9 that the realization of a simplicial
set is always a CW complex. Thus, for many of the purposes of algebraic topology, Y and
[S(Y )[ are virtually indistinguishable. So perhaps, wearing the appropriate glasses, Y and
S(Y ) can be treated as the same thing, especially if Y is a CW complex? Well return to
this idea later.
Example 4.7. As noted in Example 4.4, the realization of a simplicial set that we obtained
from a simplicial complex is the original simplicial complex. So, for example, we can obtain
a topological (n1)-sphere as the realization of the boundary of the n-simplex,
n
. What
does
n
look like as a simplicial set? Since every m-simplex of
n
is also a simplex of

n
, each can be written [i
0
, . . . , i
m
], where 0 i
0
i
m
n. The only caveat is
that we cannot allow any m-simplex that contains all of the vertices 0, . . . , n, since any such
simplex would either be the top face [0, . . . , n], itself, or a degeneration of it, and these
are not allowable faces of
n
. In summary, then, we have S
n1

= [
n
[, where
n
is the
simplicial set consisting of all sequences of the numbers 0, . . . , n that do not contain all of
the numbers 0, . . . , n.
Is this the most ecient way to realize S
n1
as the realization of a simplicial set? After all,

n
contains quite a number of simplices, many of which are nondegenerate (the interested
reader might go and count them). Here is another way to do it, at least for n 2, suggested
by CW complexes. Let X be a simplicial set whose only nondegenerate simplices are denoted
by [0] X
0
and [0, . . . , n 1] X
n1
. All simplices in X
i
, 0 < i < n 1, are the degenerate
simplices [0, . . . , 0]. This, of course, forces all of the faces of [0, . . . , n 1] to be [0, . . . , 0],
and we see that the realization [X[ is equivalent to the standard construction of S
n1
as a
CW complex by collapsing the boundary of an (n 1)-cell to a point. See Figure 18.
Figure 18: The realization of the simplicial set consisting of only two nondegenerate simplices,
one in dimension 0 and the other in dimension 2, is the sphere S
2
; this picture represents
the image of the nondegenerate simplex of dimension 2 in the realization.
The preceding example is instructive on several dierent points:
1. The second part of Example 4.7 relies strongly on the existence of degenerate simplices.
For n > 2, we cannot construct S
n1
this way as the realization of a Delta set. A Delta
22
set with an (n1)-simplex would require actual (nondegenerate) (n2)-simplices as its
faces. Of course we can still get S
n1
as the realization of the Delta set corresponding
to
n
.
2. Notice that the realization of a simplicial set does not necessarily inherit the structure
of a simplicial complex, at least not in any obvious way from the data of the simplicial
set.
3. Realizations are non-unique, in the sense that very dierent looking simplicial sets can
have the same geometric realization up to homeomorphism. This is not surprising,
since there are many ways to triangulate a piecewise-linear space.
Example 4.7 is also disconcerting in that the reader may be getting worried that realiza-
tions of simplicial sets might be very complicated to understand with all of the gluing and
collapsing that can occur. To mitigate these concerns somewhat, we rst observe that all
degenerate simplices do get collapsed down into the simplices of which they are degeneracies,
and so constructing a realization depends only on understanding what happens to the non-
degenerate simplices. A second concern would be that two nondegenerate simplices might
be glued together. This would happen if it were possible for two nondegenerate simplices
to have a common degeneracy. Luckily, this does not happen, as we demonstrate in the
following proposition. As a corollary, we can conclude that the realization of a simplicial
set is made up of the disjoint union of the interiors of the nondegenerate simplices. We
must limit this statement to the interiors as the faces of a nondegenerate simplex may be
degenerate, as in the second part of Example 4.7 - meanwhile, nondegenerate faces will look
out for themselves!
Proposition 4.8. A degenerate simplex is a degeneracy of a unique nondegenerate simplex.
In other words, if z is a degenerate simplex, then there is a unique nondegenerate simplex x
such that z = s
i
1
s
i
k
x, for some collection of degeneracy maps s
i
1
, . . . , s
i
k
.
Proof. Suppose z is a degenerate simplex. Then z = s
i
1
x
1
for some x
1
and some degeneracy
map s
i
1
. If x
1
is degenerate, we can make a similar replacement and continue inductively
until eventually we have z = s
i
1
s
i
k
x
k
for some nondegenerate x
k
. Thus z can be written
in the desired form.
Next, suppose x and y are nondegenerate simplices, possibly of dierent dimensions,
and that Sx = Ty, where S and T are compositions of degeneracy operators. Suppose
S = s
i
1
s
i
k
. Let D = d
i
k
d
i
1
. Then x = DSx = DTy, using the simplicial set axioms
for the rst equality, and, by using the simplicial set axioms to trade face maps to the right,
we obtain x =

T

Dy for some composition of face operators

D and some composition of
degeneracies

T. But, by hypothesis, x is nondegenerate, so

T must be vacuous, and we must
have x =

Dy. That is x is a face of y. But we could repeat the argument reversing x and y
to obtain that y is also face of x. But this is impossible unless x = y.
Another comforting fact is the following theorem:
23
Theorem 4.9. If X is a simplicial set, then [X[ is a CW complex with one n-cell for each
nondegenerate n-simplex of X.
Proof. We refer to Milnors paper on geometric realization [12] (or, alternatively, to [11,
Theorem 14.1]) for the proof, which is not dicult and which formalizes our discussion
preceding Proposition 4.8.
The adjointness relation. The realization functor [ [ turns out to be adjoint to the
singular simplex functor S().
Theorem 4.10. If X is a simplicial set and Y is a topological space, then
Hom
Top
([X[, Y )

= Hom
S
(X, S(Y )).
Sketch of proof. We identify the maps : Hom
Top
([X[, Y ) Hom
S
(X, S(Y )) and :
Hom
S
(X, S(Y )) Hom
Top
([X[, Y ) and leave it to the reader both to check carefully that
these are well-dened and to show that they are mutual inverses.
A map f Hom
S
(X, S(Y )) assigns to each n-simplex x X a continuous function

x
: [
n
[ Y . Let (f) be the continuous function that acts on the simplex (x, [
n
[) [X[
by applying
x
to [
n
[.
Conversely, given a function g Hom
Top
([X[, Y ), then the restriction of g to a nonde-
generate simplex (x, [
n
[) yields a continuous function [
n
[ Y and thus an element of
S(Y )
n
. If (x, [
n
[) represents a degenerate simplex, then we precompose with the appro-
priate collapse map of
n
into [X[ before applying g.
One can say much more on the relation between simplicial sets and categories of topologi-
cal spaces. For example, see Theorem 10.1 below, according to which the homotopy category
of CW complexes is equivalent to the homotopy category of simplicial sets satisfying a con-
dition called the Kan condition. The Kan condition is dened in Section 7.
5 Products
Before we move on to a look at simplicial homotopy, we will need to know about products
of simplicial sets. For those accustomed to products of simplicial complexes or products of
chain complexes, the denition of the product of simplicial sets looks surprisingly benign by
comparison.
Denition 5.1. Let X and Y be simplicial sets. Then their product X Y is dened by
1. (X Y )
n
= X
n
Y
n
= (x, y) [ x X
n
, y Y
n
,
2. if (x, y) (X Y )
n
, then d
i
(x, y) = (d
i
x, d
i
y),
3. if (x, y) (X Y )
n
, then s
i
(x, y) = (s
i
x, s
i
y).
24
Notice that there are evident projection maps
1
: X Y X and
2
: X Y Y
given by
1
(x, y) = x and
2
(x, y) = y. These maps are clearly simplicial morphisms.
Denition 5.1 looks disturbingly simple-minded, but it is vindicated by the following
important theorem.
Theorem 5.2. If X and Y are simplicial sets, then [X Y [

= [X[ [Y [ (in the category of
compactly generated Hausdor spaces). In particular, if X and Y are countable or if one of
[X[, [Y [ is locally nite as a CW complex, then [X Y [

= [X[ [Y [ as topological spaces.
We refer the reader to [11, Theorem 14.3] or [12] for a proof in the latter situations and
to [7, Chapter III] for a proof of the general case. However, since an example is perhaps
worth a thousand proofs, we will take a detailed look at some special cases.
Example 5.3. Let X be any simplicial set, and let Y =
0
= [0]. Since
0
has a unique
element in each dimension, X
0

= X. So indeed, [X
0
[

= [X[ [
0
[

= [X[.
Example 5.4. The rst interesting example is
1

1
. We would like to see that [
1

1
[

=
[
1
[ [
1
[, the square. As discussed in Section 4, we need to focus on the nondegenerate
simplices of
1

1
. The reader can refer to Figure 19 for the following discussion.
Figure 19: The realization of
1

1
First, in dimension 0, we have the product 0-simplices
X
0
= ([0], [0]), ([1], [0]), ([0], [1]), ([1], [1]),
the four vertices of the square.
In dimension 1, we have the pairs (e, f), where e and f are 1-simplices of
1
. There are
three possibilities for each of e and f - [0, 0], [0, 1], and [1, 1]. So there are nine 1-simplices
of
1

1
.
There is only one 1-simplex that is made up completely of nondegenerate simplices:
([0, 1], [0, 1]). Since d
0
([0, 1], [0, 1]) = (0, 0) and d
1
([0, 1], [0, 1]) = (1, 1), the simplex ([0, 1], [0, 1])
must be the diagonal. Those with one nondegenerate and one degenerate 1-simplex are
([0, 0], [0, 1]), ([0, 1], [0, 0]), ([1, 1], [0, 1]) and ([0, 1], [1, 1]), which, as we see by checking the
endpoints, are respectively the left, bottom, right, and top of the square. The other four
25
1-simplices are the degeneracies of the vertices. For example, ([0, 0], [1, 1]) = (s
0
[0], s
0
[1]) =
s
0
([0], [1]).
Now for the 2-simplices - heres where things get a little tricky. There are four 2-simplices
of
1
: [0, 0, 0], [0, 0, 1], [0, 1, 1], and [1, 1, 1]. So there are sixteen 2-simplices of
1

1
.
There are two possible degeneracy maps, s
0
and s
1
, from (
1

1
)
1
to (
1

1
)
2
. These act
on the nine 1-simplices, but there are not eighteen degenerate 2-simplices since s
0
s
0
= s
1
s
0
,
and we know there are four degenerate 1-simplices s
0
v
i
of
1

1
corresponding to the
degeneracies of the four vertices. Removing these redundancies leaves fourteen degenerate
simplices. There are no other redundancies since s
0
s
0
= s
1
s
0
is the only relation on s
1
and
s
0
. The remaining two 2-simplices are nondegenerate. These turn out to be ([0, 0, 1], [0, 1, 1])
and ([0, 1, 1], [0, 0, 1]), which are the two triangles, as one can check by computing face maps.
Next, we need to see that all 3-simplices of
1

1
and above are degenerate. We
rst observe that each 3-simplex of
1
must be a double degeneracy of a 1-simplex (since
there are no nondegenerate simplices of
1
of dimension greater than 1). But there are
only six such options, of the forms s
0
s
0
e, s
0
s
1
e, s
1
s
0
e, s
1
s
1
e, s
2
s
0
e and s
2
s
1
e for a (possibly
degenerate) 1-simplex e. However, the simplicial set axioms reduce this to the possibilities
s
1
s
0
e, s
2
s
0
e, and s
2
s
1
e. But then, again by the axioms,
(s
1
s
0
e, s
1
s
0
f) = s
1
(s
0
e, s
0
f)
(s
1
s
0
e, s
2
s
0
f) = (s
0
s
0
e, s
0
s
1
f) = s
0
(s
0
e, s
1
f)
(s
1
s
0
e, s
2
s
1
f) = (s
1
s
0
e, s
1
s
1
f) = s
1
(s
0
e, s
1
f)
(s
2
s
0
e, s
1
s
0
f) = (s
0
s
1
e, s
0
s
0
f) = s
0
(s
1
e, s
0
f)
(s
2
s
0
e, s
2
s
0
f) = s
2
(s
0
e, s
0
f)
(s
2
s
0
e, s
2
s
1
f) = s
2
(s
0
e, s
1
f)
(s
2
s
1
e, s
1
s
0
f) = (s
1
s
1
e, s
1
s
0
f) = s
1
(s
1
e, s
0
f)
(s
2
s
1
e, s
2
s
0
f) = s
2
(s
1
e, s
0
f)
(s
2
s
1
e, s
2
s
1
f) = s
2
(s
1
e, s
1
f).
So all 3-simplices of
1

1
are degenerate. It also follows that all higher dimension
simplices are degenerate: the terms in any such product must be further degeneracies of
these particular doubly degenerate 1-simplices, and using the simplicial set axioms, we can
move s
0
and s
1
to the left in all expressions. Then we can proceed as in the above list of
computations.
That last bit isnt very intuitive, but the low-dimensional part makes some sense. If
we take the product of two CW complexes, the cells of the product will be the products
of the cells C
1
C
2
, where C
1
and C
2
are not necessarily of the same dimension. These
mixed dimensional cells occur here as products of nondegenerate simplices with degenerate
simplices. What makes matters dicult is that we must preserve a simplicial structure. This
forced triangulation is what makes matters somewhat complicated.
It will also be useful for us to look more closely at the products
p

q
. After all,
all products will be made up of these building blocks. The main point of interest for us is
26
that the simplicial product construction yields the same triangulation structure that may be
familiar from homotopy arguments in courses in beginning algebraic topology.
Example 5.5. Since we know that [
p

q
[ = [
p
[ [
q
[, let us focus on the nondegenerate
(p +q)-simplices of
p

q
. We let E
j
stand for the unique nondegenerate j-simplex of
j
.
We note immediately that any nondegenerate (p+q)-simplex s of
p

q
(and hence the only
ones that appear nondegenerately in the realization) must have the form s = (SE
p
, S

E
q
),
where S and S

are sequences of degeneracy maps. Why? Otherwise s would have to be of

the form s = (

St,

S

), where

S and

S

are again sequences of degeneracy maps and t and t

are faces of E
p
and E
q
, respectively. But in this case, we would have s F F

, where F
and F

are the simplicial subsets corresponding to proper faces of

p
and
q
. Consequently
the image of s [
p+q
[ in the realization of
p

q
will in fact lie within the realization
[F[[F

[. But then dim[s[ < p+q, so s could not have been a nondegenerate (p+q)-simplex.
So now we see that s = (SE
p
, S

E
q
), and for dimensional reasons, we can write this as
s = (s
iq
s
i
1
E
p
, s
jp
s
j
1
E
q
). Furthermore, using the simplicial set axioms, we can assume
that 0 i
1
< < i
q
< p +q and 0 j
1
< < j
q
< p +q. Now notice that the collection
i
1
, . . . , i
q
, j
1
, . . . , j
p
consists of p + q numbers from 0 to p + q 1. Furthermore, there can
be no redundancy, since if i
k
= j
k
for some k and k

, then again by the axioms, we can

pull these indices to the front to get s = (s
i
k

SE
p
, s
j
k

E
q
) = s
i
k
(

SE
p
,

S

E
q
) for some

S,

S

,
making s degenerate.
Thus we conclude that the nondegenerate (p + q)-simplices of
p

q
are precisely
those of the form s = (s
iq
s
i
1
E
p
, s
jp
s
j
1
E
q
), where the i
k
and j
k
are increasing series of
integers from 0 to p + q 1, all completely distinct.
In the special case
p

1
=
p
I, this rule for nondegenerate (p + 1)-dimensional
simplices reduces to the form s = (s
i
E
p
, s
jp
s
j
1
e), where e is the edge [0, 1] of I, and the
sequence j
1
, . . . , j
p
is increasing from 0 to p, omitting only i. Thus there are precisely p + 1
nondegenerate (p + 1)-simplices. Since e = [0, 1], notice that all of the degeneracy maps
before the gap at i must adjoin another 0 and all of those after the gap adjoin more 1s.
Thus we can also label these nondegenerate (p +1)-simplices exactly by the p +1 sequences
of length p + 2 of the form [0, . . . , 0, 1, . . . , 1] that must start with a 0 and end with a 1.
If this looks familiar, its because the standard way to triangulate the product prism
p
I
when studying simplicial homology theory is by the (p + 1)-simplices [0, . . . , k, k

, . . . , p

],
where the unprimed numbers represent vertices in
p
0 and the primed numbers represent
vertices in
p
1. The simplex [0, . . . , k, k

, . . . , p

] corresponds to k +1 zeros and p k +1

ones. See Figure 20.
For our upcoming discussion of simplicial homotopy, its also worth looking at how these
simplices are joined together along their boundaries. Lets rst look from the point of view of
writing the (p+1)-simplices of
p
I in the form S
k
= [0, . . . , k, k

, . . . , p

], where 0 k p.
If i < k, then d
i
S
k
= [0, . . . , i 1, i + 1, . . . , k, k

, . . . , p

]. But this can be thought of as a

p-simplex of [0, . . . , i 1, i + 1, . . . , p] I and so is part of the boundary
p
I. Similar
considerations hold if i > k + 1. The interesting interior cases are
d
k
S
k
= [0, . . . , k 1, k

, . . . , p

]
d
k+1
S
k
= [0, . . . , k, (k + 1)

, . . . , p

].
27
Figure 20: The realization of [
2

1
[ with nondegenerate 3-simplices [0, 1, 2, 2

], [0, 1, 1

, 2

],
and [0, 0

, 1

, 2

]
To understand the assembly of the prism
p
I, notice that d
k
S
k
= d
k
S
k1
for k > 0 and
d
k+1
S
k
= d
k+1
S
k+1
for k < p. This tells us how to glue the (p+1)-simplices together to form
[
p
I[.
In our other notation, if we have S
k
= (s
k
E
p
, s
p
s
k+1
s
k1
s
0
e), then for i < k we
have, using the axioms,
d
i
S
k
= (s
k1
d
i
E
p
, s
p1
s
k+1
s
k1
s
i
(d
i
s
i
)s
i1
s
0
e) = (s
k1
d
i
E
p
, s
p1
s
k
s
k2
s
0
e).
Notice that we use the axioms to pass d
i
through, converting each s
j
to s
j1
along the
way, until it annihilates with the original s
i
(leaving the previous s
i+1
converted to the
new s
i
). We wind up with a p-simplex that is recognizable as a p-simplex in d
i
E
p
I.
Similarly, for i > k + 1, we get d
i
S
k
= (s
k
d
i1
E
p
, s
p1
s
k
s
k2
s
0
e). The two interior
cases correspond to d
k
S
k
and d
k+1
S
k
:
d
k
S
k
= (d
k
s
k
E
p
, s
p1
s
k+1
s
k2
s
0
e) = (E
p
, s
p1
s
k+1
s
k2
s
0
e)
d
k+1
S
k
= (d
k+1
s
k
E
p
, s
p1
s
k+2
s
k1
s
0
e) = (E
p
, s
p1
s
k+2
s
k1
s
0
e).
These are not in
p
I. However, we do again see that d
k
S
k
= d
k
S
k1
for k > 0 and
d
k+1
S
k
= d
k+1
S
k+1
for k < p.
6 Simplicial objects in other categories
Before moving on to discuss simplicial homotopy, we pause to note that the categorical
denition of simplicial sets suggests a sweeping generalization.
28
Denition 6.1. Let Cat be a category. A simplicial object in Cat is a contravariant functor
X : Cat (equivalently, a covariant functor X :
op
Cat). A morphism of simplicial
objects in Cat is a natural transformation of such functors.
Another common notation, when Cat is a familiar category with objects of a given
type, is to refer to a simplicial object in Cat as a simplicial [insert type of object]. In
other words, when Cat is the category of groups and group homomorphisms, we speak of
simplicial groups. This is consistent with referring to a simplicial object in the category Set
as a simplicial set. One also commonly encounters simplicial R-modules, simplicial spaces,
and even simplicial categories!
Example 6.2. Lets unwind the denition in the case of simplicial groups. By denition, a
simplicial group ( consists of a sequence of groups (
n
and collections of group homomor-
phisms d
i
: (
n
(
n1
and s
i
: (
n
(
n+1
, 0 i n, that satisfy the axioms (1).
At this point, unfortunately, trying to picture group elements as simplices breaks down a
little bit since there is so much extra structure around (what does it mean geometrically to
multiply two simplices?). Nonetheless, it is still helpful to refer mentally to the category , in
which we can visualize each simplex [n] as representing a group and picture movement toward
each n 1 face as representing a dierent group homomorphism to the group represented
by [n 1]. See Figure 21.
Figure 21: A pictorial representation of a 2-simplex of a simplicial group with arrows rep-
resenting the face morphisms from dimension 2 to dimension 1 and from dimension 1 to
dimension 0
Example 6.3. Suppose X is a simplicial set. Then we can form the simplicial group C

(X)
with (C

X)
n
= C
n
(X) dened to be the free abelian group generated by the elements of X
n
with d
i
in C

(X) taken to be the linear extensions of the face maps d

i
of X. We can also
form the total face map d =
n

i=0
(1)
i
d
i
: C
n
(X) C
n1
(X) and then dene the homology
H

(X) as the homology of this chain complex.

If X = S(Y ), the singular set as dened in Example 3.7, then we have H

(X) = H

(Y ),
the singular homology of the space Y .
29
Example 6.4. Heres an example of a simplicial group that is important in the theory of
group cohomology. Let G be a group, and let BG be the simplicial group dened as follows.
Let BG
n
= G
n
, the product of G with itself n times. G
0
is just the trivial group e. For
an element (g
1
, . . . , g
n
) BG
n
, let
d
0
(g
1
, . . . , g
n
) = (g
2
, . . . , g
n
)
d
i
(g
1
, . . . , g
n
) = (g
1
, . . . , g
i
g
i+1
, . . . g
n
) if 0 < i < n
d
n
(g
1
, . . . , g
n
) = (g
1
, . . . , g
n1
)
s
i
(g
1
, . . . , g
n
) = (g
1
, . . . , g
i
, e, g
i+1
, . . . , g
n
).
The reader can check that this denes a simplicial group. The realization of the under-
lying simplicial set turns out to be the classifying space of the group G. For more on this
simplicial group and its uses, the reader may consult [21, Chapter 8].
7 Kan complexes
One of the goals of the development of simplicial sets (and other simplicial objects) was
to nd a combinatorial way to study homotopy theory, just as simplicial homology theory
allows us to derive invariants of simplicial complexes in a purely combinatorial manner (at
least in principle). Unfortunately, it turns out that not all simplicial sets are created equal
as regards their usefulness toward this goal. The underlying reason turns out to be (once
again, at least in principle) related to the reason that homotopy theorists prefer to work
with CW complexes and not arbitrary topological spaces. Pairs of CW complexes satisfy
the homotopy extension property, i.e. inclusions of subcomplexes are cobrations (see, e.g.,
[4]). The condition we need to impose on simplicial sets to make them appropriate for
the study of homotopy is similarly an extension condition. When seen through suciently
advanced lenses, such as from the model category viewpoint presented in [8], the extension
condition on simplicial sets and the homotopy extension property in topology are essentially
equivalent.
As with much else in the theory of simplicial sets, the extension condition comes from a
fairly straightforward idea that is often completely obfuscated in the formal denition.
To explain the idea, we rst need the following denition.
Denition 7.1. As a simplicial complex, the kth horn [
n
k
[ on the n-simplex [
n
[ is the
subcomplex of [
n
[ obtained by removing the interior of [
n
[ and the interior of the face
d
k

n
. See Figure 22. We let
n
k
refer to the associated simplicial set. This simplicial set
consists of simplices [i
0
, . . . , i
m
] with 0 i
0
i
m
n such that 1) not all numbers
0, . . . , n are represented (this would be the top face or a degeneracy thereof) and 2) we never
have all numbers except k represented (this would be the missing (n1)-face or a degeneracy
thereof).
The extension condition, also known as the Kan condition (after Daniel Kan), says that
whenever we see a horn on an n-simplex within a simplicial set, the rest of the simplex is
there, too. Heres an elegant way to say this:
30
Figure 22: The three horns on [
2
[
Denition 7.2. The simplicial object X satises the extension condition or Kan condition if
any morphism of simplicial sets
n
k
X can be extended to a simplicial morphism
n
X.
Such an X is often called a Kan complex or, in more modern language, is referred
to as being brant (note the risk of confusion here between simplicial sets and simplicial
complexes).
We next present an equivalent formulation that is often used. This version has its ad-
vantages from the point of view of conciseness of combinatorial information, but it is much
less conceptual.
Denition 7.3 (Alternate version of the Kan condition). The simplicial set X satises the
Kan condition if for any collection of (n 1)-simplices x
0
, . . . , x
k1
, x
k+1
, . . . , x
n
in X such
that d
i
x
j
= d
j1
x
i
for any i < j with i ,= k and j ,= k, there is an n-simplex x in X such
that d
i
x = x
i
for all i ,= k.
The condition on the simplices x
i
of the alternative denition glues them together to
form the horn
n
k
, possibly with degenerate faces, within X, and the denition says that we
can extend this horn to a (possibly degenerate) n-simplex in X.
Example 7.4. Not even the standard simplices
n
, n > 0, satisfy the Kan condition! Let

1
= [0, 1] be the standard 1-simplex, and consider the horn
2
0
, which consists of the edges
[0, 2] and [0, 1] of
2
, along with their degeneracies. Now consider the simplicial morphism
that takes [0, 2]
2
0
to [0, 0]
1
and [0, 1]
2
0
to [0, 1]
1
. There is a unique such
simplicial map since weve specied what happens on all the nondegenerate simplices of
2
0
.
Notice that this is perfectly well-dened as a simplicial map since all functions on all simplices
are order-preserving. However, this cannot be extended to a map
2

1
since we have
already prescribed that 0 0, 1 1, and 2 0, which is clearly not order-preserving on

2
.
Example 7.5. It is easy to check that
0
does satisfy the Kan condition.
Example 7.6. Given a topological space Y , the simplicial set S(Y ) does satisfy the Kan
extension condition. It is actually fairly straightforward to see this. Consider any morphism
of simplicial sets f :
n
k
S(Y ). This is the same as specifying for each n 1 face, d
i

n
,
i ,= k, of
n
a singular simplex
i
: [
n1
[ Y . Every other simplex of
n
k
is a face
or a degeneracy of a face of one of these (n 1)-simplices, and so the rest of the map
f is determined by this data. Furthermore, the compatibility conditions coming from the
simplicial set axioms ensure that the topological maps
i
piece together to yield, collectively,
31
a continuous function f : [
n
k
[ Y . It is now a simple matter to extend this function
to all of [
n
[: let : [
n
[ [
n
k
[ be any continuous retraction (which certainly exists:
([
n
[, [
k
n
[) is homeomorphic to (I
n1
I, I
n1
0)), and dene = f : [
n
[ Y . This
is a singular n-simplex whose faces d
i
f, i ,= k, are precisely the singular simplices
i
. Thus
this determines the desired extension. See Figure 23.
Figure 23: A demonstration that the singular set satises the Kan condition
Example 7.7. Any simplicial group is also, by neglect of structure, a simplicial set. All such
simplicial sets arising from simplicial groups satisfy the Kan condition. The proof is not
dicult, but I dont know of a version that is particularly illuminating. Since we will not
have much further use for this fact in these notes, we refer the reader to [13, Theorem 2.2]
for a proof.
8 Simplicial homotopy
In this section we begin to look at the homotopy properties of simplicial sets. This is one of
the key reasons that the theory of simplicial sets exists - to allow us to turn homotopy theo-
retic problems, at least in principle, into combinatorial problems by studying the homotopy
groups of simplicial sets instead of those of topological spaces. In order to get started with
simplicial homotopy, it is necessary to restrict attention to simplicial sets satisfying the Kan
condition. This is not as large a handicap as it rst appears, however, since we have already
seen that, given a topological space Y , the singular set S(Y ) satises the Kan condition,
and eventually, we will see that S(Y ) constitutes an appropriate combinatorial stand-in for
Y .
As we proceed, the reader should bear in mind the extent to which many of the ideas
and denitions mirror those in topological homotopy theory. This may prove a comfort (or
cause serious worry!) at those junctures where the mirror appears somewhat warped by the
combinatorial complexity of the simplicial version.
We begin, naturally enough, with
0
, corresponding to the homotopy relationship between
maps of points. This is a quite tractable warm-up for what is to follow.
32
8.1 Paths and path components
As in topology, when talking about homotopy, we will let I stand for the simplicial set

1
= [0, 1]. As a simplicial set, I has the nondegenerate 1-simplex [0, 1], the nondegenerate
0-simplices [0] and [1], and all other simplices are degenerate. Each simplex has the form
[0, . . . , 0, 1, . . . , 1] (possibly with no 0s or no 1s).
Denition 8.1. A path in a simplicial set X is a simplicial morphism p : I X. Equiva-
lently, a path in X is a 1-simplex p X
1
. If p is a path in X, d
1
p = p[0] is called the initial
point of the path and d
0
p = p[1] is called the nal point or terminal point.
Denition 8.2. Two 0-simplices a and b of the simplicial set X are said to be in the same
path component of X if there is a path p with initial point a and nal point b.
Already this denition appears slightly odd if youre used to working with simplicial
complexes. In a connected simplicial complex, one might have to traverse several edges to
link two vertices. Here we require it to be done all with one edge. Furthermore, we would
expect being in the same path component to be an equivalence relation. This is not at
all clear, say, in an ordered simplicial complex in which we can have a < b or b < a but not
both. What rescues this denition is the Kan condition.
Theorem 8.3. If X is a Kan complex, then being in the same path component is an
equivalence relation.
Proof. We will go through the proof in detail as it is very illuminating of how to think
geometrically about simplicial sets.
Reexivity. This one is easy: for any vertex [a], s
0
[a] is a path from a to a.
Transitivity. Consider
2
= [0, 1, 2]. If p
1
is a path from a to b and p
2
is a path from b
to c, then let f :
2
1
X take [0, 1] to p
1
and [1, 2] to p
2
. See Figure 24. The Kan condition
lets us extend f to

f :
2
X, and

f[0, 2] gives us a path from a to c.
Figure 24: The transitivity relation on path connectedness via the Kan condition
33
Symmetry. This is only slightly more tricky than the transitivity condition. See Figure
25. Let p be a path in X from a to b. We need a path the other way. Think of p as the [0, 1]
side of
2
. Let the [0, 2] side of
2
represent s
0
[a], which must exist since X is a simplicial
set. Notice that d
0
s
0
[a] = d
1
s
0
[a] = [a]. At this point, we can label the three vertices [0, 1, 2]
of
2
as [a, b, a], and we have a simplicial map on
2
0
taking [0, 1] to p and [0, 2] to s
0
[a].
The Kan condition tells us that this map can be extended to all of
2
and [1, 2] gets taken
to a path p from b to a.
Figure 25: The symmetry relation on path connectedness
Notice how important the Kan condition is here.
Since we have demonstrated that being in the same path component is an equivalence
relation, we have equivalence classes.
Denition 8.4. We denote the set of path components of X (i.e. the equivalence classes of
vertices of X under the relation of being in the same path component) by
0
(X).
So far, this is comfortingly familiar.
8.2 Homotopies of maps
There are at least two classical versions of the denition of simplicial homotopy, and at least
two more modern versions for which we refer the reader to [8]. Of the two classical versions,
one has the expected form for a homotopy, H : XI Y . The other is more closely related
to the homotopies we see in chain complexes

H : X
n
Y
n+1
. We will look at both of these
and see how they are related.
Perhaps the most natural denition of simplicial homotopy looks something like this:
Denition 8.5 (Simplicial homotopy 1). Two simplicial maps f, g : X Y are homotopic
if there is a simplicial map H : X I Y such that H[
X0
= g and H[
X1
= f (i.e., if
g = Hi
0
and f = Hi
1
, where i
0
, i
1
are the evident simplicial inclusion maps i
0
: X[0]
X I and i
1
: X [1] X I).
Unfortunately, here is the denition of simplicial homotopy one nds quite often in the
literature:
Denition 8.6 (Simplicial homotopy 2). Two simplicial maps f, g : X Y are homotopic
if for each p there exist functions h
i
= h
p
i
: X
p
Y
p+1
for each i, 0 i p, such that
34
1.
d
0
h
0
= f
d
p+1
h
p
= g
2.
d
i
h
j
= h
j1
d
i
if i < j
d
j+1
h
j+1
= d
j+1
h
j
d
i
h
j
= h
j
d
i1
if i > j + 1
3.
s
i
h
j
= h
j+1
s
i
if i j
s
i
h
j
= h
j
s
i1
if i > j.
It will take some doing to see how these two denitions are related. This was one of the
initial motivations for writing this exposition!
As usual, we will consider the universal example, X =
p
, since once we understand how
a homotopy works on a single simplex, we will also understand what happens along its faces
and degeneracies, and everything else is determined by how the simplices are glued together.
The key here is to recall Example 5.5 of Section 5, in which we showed how the prism
[
p
I[ is decomposed into simplices. In particular, it consists of p + 1 nondegenerate
(p +1)-simplices that we labeled S
k
(
p
I)
p+1
, 0 k p. Suppose now that we have a
homotopy H :
p
I Y from f to g. Everything is determined by what H does to the
S
k
, since every other nondegenerate simplex in
p
I is a face of one of these simplices. All
other simplices in
p
I are degenerate, and so their images are determined by the images
of the simplices of which they are degeneracies.
How does this relate to the combinatorial Denition 8.6? Let us denote the unique
nondegenerate p-simplex of
p
by E
p
. In this version, there are p+1 functions h
i
: E
p
Y
p+1
.
Each of the p + 1 functions h
i
assigns to E
p
a (p + 1)-simplex of Y . Collectively, these give
us the image of the prism over E
p
in Y .
To see this, we use the notation S
k
= [0, . . . , k, k

, . . . , p

], 0 k p, for the (p + 1)-

simplices of the prism
p
I (see Example 5.5). Given H :
p
I Y , let h
k
(E
p
)
correspond to the image H(S
k
) in Y . Now lets look at the conditions in Denition 8.6 and
see what they mean.
Starting with the rst conditions, d
0
h
0
(E
p
) = d
0
H(S
0
) = H(d
0
S
0
) = H(d
0
[0, 0

, . . . , p

]) =
H([0

, . . . , p

]) = H i
1
(E
p
) = f(E
p
), using the rst denition of homotopy. Similarly,
d
p+1
h
p
(E
p
) = H([0, . . . , p]) = H i
0
(E
p
) = g(E
p
). So these conditions assure that the ends
of the prism really are controlled by the maps f and g.
The rst and third equations of the second set of conditions mirror the observations made
in Example 5.5 that most of the boundaries of the (p +1)-simplices of the prism
p
I are
themselves simplices of the prisms built on the boundary faces of
p
. So these equations
35
ensure that these faces of the h
i
(
p
) are compatible with the actions of the homotopy maps
h
j
i
of lower dimensions j < p on the faces of
p
. The second equation is the condition that
the neighboring simplices S
k
and S
k+1
share a boundary. We invite the reader to glean these
combinatorial details from the calculations in Example 5.5.
The third set of equations can also be obtained in a fairly straightforward manner by
working with the S
k
. For example, we observe that for i j, s
i
S
j
= [0, . . . , i, i, . . . , j, j

, . . . , p

],
which is also the (j +1)st prism simplex on the degenerate simplex [0, . . . , i, i, . . . , j 1, j, j +
1, . . . , p]. In other words, the ith degeneracy of the jth prism (p + 1)-simplex over
p
is
the (j + 1)st prism simplex over the ith degeneracy of
p
. The geometric idea is a bit less
obvious than in the preceding paragraphs, but really this is just the condition that the way
the homotopy acts on degenerate simplices is determined by how it acts on the simplices of
which they are degeneracies.
Having described how the combinatorial conditions of the second denition correspond
to the more geometric ideas of the rst denition, we now leave it to the interested reader to
verify the complete equivalence of the two denitions, in particular to verify that the data
given by all the h
j
i
is enough to reconstruct H.
We would like homotopy to be an equivalence relation, but this will not hold in general.
For example, in our discussion of path connectedness, which we see in the current language
corresponds directly to homotopies of maps
0
X, we saw that path connectedness is not
always an equivalence relation. However, the discussion of path connectedness might lead
one to suspect that we will be safe in the world of Kan complexes, and this is so.
Theorem 8.7. Homotopy of maps X Y is an equivalence relation if Y is a Kan complex.
If f and g are homotopic, we denote that by f g.
We invite the reader to prove this by extending the argument given for path connected-
ness.
It is also fairly straightforward to verify other expected elementary fact about homotopy;
for instance if f f

, then fg f

g and gf gf

. Also, homotopic maps induce the same

homomorphisms on homology groups (see Section 3 - this follows as for the usual proof in
singular homology theory by using the triangulation of the homotopy prism; see, e.g. [14]).
8.3 Relative homotopy
The notions of subcomplexes and relative homotopy oer no surprises, but we record the
denitions for clarity.
Denition 8.8. If X is a simplicial set, then A is a simplicial subset of X, denoted A < X,
if A itself is a simplicial set such that A
n
X
n
for all n and the face and degeneracy maps of
A agree with those from X. A pair of simplicial sets is often denoted by (X, A). Simplicial
maps of pairs (X, A) (Y, B) are simplicial maps X Y such that the image of A is
contained in B.
Denition 8.9. If (X, A) are a simplicial set and simplicial subset and both X and A satisfy
the Kan condition, then (X, A) is called a Kan pair.
36
Example 8.10. An important example of a simplicial subset of a simplicial set X is a basepoint
for X, consisting of an element of X
0
and all of its degeneracies. We will denote basepoints
by . Notice that is isomorphic as a simplicial set to
0
and can be considered as an image

0
X of a simplicial map. Since
0
is a Kan complex, (X, ) will be a Kan pair if X is
Kan.
Example 8.11. Note that it is not automatic that a subcomplex of a Kan complex be Kan.
For instance, we know from Example 7.4 that the simplex
1
is not a Kan complex. We
also know that the singular set S([
1
[) on the space [
1
[ is a Kan complex, by Example
7.6. But the former is a subcomplex of the latter, realized by the singular simplices that
represent [
1
[ as a simplicial complex. Namely,
1
corresponds to the subcomplex of o([
1
[)
generated by the singular 0-simplices
0
: [
0
[ [0] and
1
: [
0
[ [1], by the singular
1-simplex id : [
1
[ [
1
[, and by their degeneracies.
Denition 8.12. A homotopy H : X I Y is a homotopy rel A if the restriction of H
to A I can be factored as H[
AI
= g
1
: A I Y , where g is a simplicial map A Y
and
1
is the projection A I A. If Y is Kan, then homotopy rel A is an equivalence
relation.
While considering simplicial pairs, there is another crucial theorem we should mention:
the homotopy extension theorem for simplicial maps to Kan complexes:
Theorem 8.13 (Homotopy extension theorem). Let (X, A) be a pair of simplicial sets and
Y a Kan complex. Suppose there is a simplicial map f : X Y and a simplicial homotopy
H : AI Y such that H[
A0
= f[A. Then there exists an extension F : X I Y such
that F[
AI
= H and F[
X0
= f.
Unfortunately, the proofs I know would all take us too far aeld, so we refer the reader
to [13, Chapter 1, Appendix A] for a combinatorial treatment or [8, Section I.4] for a more
modern treatment.
9
n
(X, )
In this section, we will discuss the homotopy groups of Kan complexes. This section is a
bit more technical than the preceding ones, as we here need some theorems and not just
denitions. This section should serve as good technical practice for the reader preparing to
go on to read further material on simplicial objects.
Given a Kan complex with basepoint (X, ), there are at least four ways to dene

n
(X, ):
1. One can dene these groups directly as homotopy classes of maps of (simplicial) spheres
to X.
2. There is a purely combinatorial denition.
37
3. As in algebraic topology, one can rst dene appropriate iterated simplicial loop spaces

n
(X) and dene
n
(X) =
0
(
n
(X)).
4. As a more topological alternative, one could try the topological homotopy groups of
the realization of X, i.e.
n
([X[, [ [).
We will focus on the relationship between the rst two of these, referring the interested
reader to [13] for the third approach. For hints at the relevance of the fourth approach, see
Theorem 10.1, below, as well as the discussion in Section 10 in general.
The denition of
n
(X, ) in terms of spheres is straightforward once we decide what a
sphere is. Example 4.7 teaches us that there is more than one reasonable denition, or at
least more than one simplicial set whose realization is a sphere. In fact, we will see that
both versions treated in that example are acceptable.
Denition 9.1 (First denition of
n
). Given a Kan complex with basepoint (X, ), dene

n
(X, ), n > 0, to be the set of homotopy equivalence classes of maps (
n+1
, ) (X, ).
Here, we take for the basepoint of
n+1
the simplicial subset of
n+1
generated by the
vertex [0], and all homotopies are relative to the basepoint.
The requirement that X be Kan is necessary for homotopy to be an equivalence relation.
Of course, we want
n
(X, ) to be a group, but this will have to wait a moment. Lets rst
work toward the more combinatorial denition. This takes a little bit of preliminary eort.
Denition 9.2. We say that two n-simplices x, x

X
n
are homotopic if
1. d
i
x = d
i
x

for 0 i n, and
2. there exists a simplex y X
n+1
such that
(a) d
n
y = x,
(b) d
n+1
y = x

, and
(c) d
i
y = s
n1
d
i
x = s
n1
d
i
x

, 0 i n 1.
The idea here is that x and x

have the same boundaries and that y provides the homotopy

between them, rel boundary, by letting x and x

be two of the faces of y, while the rest of

the faces of y degenerate to the most straightforward degeneracies of the boundaries of x
and y. See Figure 26.
It can be shown directly that homotopy of simplices is an equivalence relation if X is a
Kan complex. The argument is a generalization of the one showing that path connectedness
is an equivalence relation. Again the idea is to arrange a simplex so that the pieces we know
fall on certain faces of horns and the pieces wed like to show exist fall on the missing faces.
Then these relations must exist due to the Kan extension condition. We refer the interested
reader to [11, Section I.3].
Denition 9.3 (Second denition of
n
). Given a Kan complex with basepoint (X, ), we
can also dene
n
(X, ), n > 0, as the set of equivalence classes of n-simplices x X
n
with
d
i
x for all i, 0 i n, up to homotopy of simplices.
38
Figure 26: Above: a homotopy of 1-simplices. Below: a homotopy of 2-simplices. The
picture in the bottom right depicts two 2-simplices glued together along their boundaries.
This version of the homotopy groups corresponds more closely to our second version of
the sphere in Example 4.7. Recall that, as a simplicial set, this version of the sphere S
n
had
only two nondegenerate simplices: one in dimension n and one in dimension 0. An n-simplex
of X all of whose faces live in can be thought of as the image of that simplicial version of
S
n
in X. Thus this denition of
n
(X, ) also makes some geometric sense. However, there
are some obvious questions, such as: Why do the rst and second denitions of
n
agree?
And where is the group structure we expect?
To answer the rst question, we need a series of lemmas:
Lemma 9.4. If X is Kan and d
i
x = d
i
x

for all i, we obtain the same equivalence relation

as in Denition 9.2 if we instead require that d
r
y = x, d
r+1
y = x

for some 0 r n, and

d
i
y = d
i
s
r
x = d
i
s
r
x

for i ,= r, r + 1.
Proof. We refer the reader to [11] for the full proof, which is contained within Lemma 5.5
there. The idea is to show that the case of the denition using r, r + 1 is equivalent to the
version with r + 1, r + 2 for each relevant r. This is done using an extension argument by
which one creates an (n + 2)-simplex which has the two desired homotopies on two of the
sides. We illustrate a low-dimensional case in Figure 27: Suppose that x, x

are 1-simplices
and that we have a y with d
0
y = x, d
1
y = x

. We want to nd a z with d
1
z = x, d
2
z = x

.
We form the horn
2
0
, shown attened on the right of Figure 27. We embed y as [0, 1, 3] (note
that this maintains its orientation simplicially despite the oddities of the drawing). We let
the other sides of the horn be appropriate degeneracies of x

. Notice that there is no trouble

embedding this horn in X extending y X. Now the Kan condition assures us that we can
extend this embedding to all of
3
, including the remaining face [1, 2, 3]. We can check that
this last face can be taken as the desired z (be careful to notice that d
1
[1, 2, 3] = [1, 3] and
d
2
[1, 2, 3] = [1, 2]).
39
The idea in higher dimensions is precisely the same; the extra faces of the horn that
exist in higher dimensions contain other degeneracies of faces of x - see [11, Lemma 5.5], [3,
Proposition 1.19].
Figure 27: Shifting indices in the homotopy relation. Here represents d
2
s
0
x = d
2
s
0
x

,
which is a degenerate 1-simplex, both vertices being the rst vertex of x, which is also the
rst vertex of x

.
Lemma 9.5. If X is Kan, two n-simplices x, x

X are homotopic in the sense of Denition

9.2 if and only if the inclusion maps f :
n
X and f

:
n
X that represent x and x

are homotopic rel boundary as maps.

Proof. Of course to say that f represents x means that f takes the nondegenerate n-simplex
E
n
of
n
to x X.
One direction of the argument is fairly straightforward. In order to show that f and f

are homotopic, it suces to nd a chain of n + 1 simplices of dimension n + 1, representing

the images of nondegenerate simplices of the prism
n
I, such that the top and bottom
faces of the rst and last simplex represents x and x

. But if we know that x and x

are
homotopic as simplices, we know there is one (n + 1)-simplex y connecting them with, say,
d
n
y = x, d
n+1
y = x

, and d
i
y for all other i. So now we just let y be the (n + 1)st
simplex h
n
(
n
), and we let h
i
(
n
) = s
i
x for 0 i n. In other words, we let the last
nondegenerate simplex in
n
I do the work of the homotopy, and we just collapse all the
rest into the face representing x. See Figure 28.
In the other direction, suppose we have an actual homotopy rel from x to x

thought of
as inclusion maps. By denition, this gives us a prism
n

1
X whose top is x and whose
bottom is x

. We know from the discussion in Example 5.5 that each of the nondegenerate
(n +1)-simplices of the prism has two n-faces that are not in
n

1
, and the rest are in

n

0
, all of which goes to in X. Furthermore, it is not hard to check that the two
n-faces not in
n

1
are consecutive faces. In particular, using the notation of Example
5.5, these faces are d
k
S
k
and d
k+1
S
k
. Thus by Lemma 9.4, each S
k
is a homotopy between
these two faces. Since the top and bottom faces of the prism are x and x

, we obtain a
simplicial homotopy between x and x

using the transitivity of simplicial homotopy.

Thus, to show that our two denitions of
n
(X, ) agree, it is only necessary to prove
the following lemma, which is familiar in the context of algebraic topology. The proof is
40
Figure 28: We label the vertices with the prism notation of Example 5.5. The bottom
simplex y is a homotopy of the simplices x and x

. Adjoining the degenerate simplex s

0
x
shows how to obtain a model prism for the homotopy from x to x

as inclusion maps.
somewhat long, but we provide most of the details, as it is dicult to nd a direct proof in
the standard expositions.
Lemma 9.6. If X is a Kan complex, there is a bijection between homotopy classes of maps
f : (
n+1
, ) (X, ) and homotopy classes of maps g : (
n
,
n
) (X, ).
Proof. Given g : (
n
,
n
) (X, ), it is easy to construct an associated f by identifying

n
with d
0

n+1
. Then we let f : (
n+1
, ) (X, ) be dened so that f is given by g on
d
0

n+1
and by the unique map to on each d
i

n+1
, i > 0. It is also straightforward to see
that any homotopy of g rel
n
determines a homotopy of f rel .
Conversely, suppose we are given f : (
n+1
, [0]) (X, ). We show that f is homotopic
to a function

f that takes
n+1
0
to . Then we can let g be f[
d
0

n+1.
We rst observe, as noted in the proof of Lemma 9.5, that to construct a homotopy
between two inclusions of k-simplices x and x

in X, it suces to nd a simplex y in X with

d
k
y = x, d
k+1
y = x

since this can be considered one of the blocks of a prism, and the rest
of the prism can be lled up with degeneracies of x or x

.
Keeping this in mind, we proceed by induction with the following induction step: Suppose
f
k1
:
n+1
X is such that f([0]) and f(z) for all simplices z
n+1
of
dimension k 1 such that [0] is a vertex of z, then there is a homotopy from f
k1
to an
f
k
that takes all simplices up to dimension k having [0] as a vertex to . Furthermore, the
homotopy can be performed rel the faces of dimension k 1 having [0] as a simplex
Clearly we can take f
0
= f. So suppose we have constructed f
k1
for k 1. We need
only nd the desired homotopy on the k-simplices of
n+1
that have [0] as a vertex, and then
we can apply the homotopy extension theorem, Theorem 8.13. So let z be a k-simplex of

n+1
with 0 as a vertex. We know that f
k1
(d
i
z) for i ,= 0. Now, consider the horn
k+1
0
,
and note that we can map
k+1
0
into X such that the k-face corresponding to d
k+1

k+1
is
f
k1
(z) and such that all other k-faces are taken into . Notice that this is possible precisely
because f
k1
(d
i
z) for i ,= 0. Now since X is a Kan complex, we can extend this horn to
41
a (k + 1)-simplex y in X such that d
k+1
y = f
k1
(z) and d
k
y . As noted, this is enough
to construct a homotopy on z from f
k1
(z) to the unique map of z into . In addition, this
is a homotopy rel those faces of z that have [0] as a simplex. Notice also that it is possible
to nd such homotopies for all such z independently and compatibly. In this way, we get
a homotopy on the k-simplices of
n+1
having [0] as a vertex from f
k1
to the map to .
Extending this homotopy by the homotopy extension theorem yields the desired homotopy
to f
k
.
Continuing inductively, we obtain a function f
n+1
:
n+1
X homotopic to f such
that
n+1
0
is taken to . Now we can dene g to be the restriction of f
n+1
to d
0

n+1
.
If f, f

: (
n+1
, [0]) (X, ) are homotopic rel [0], then we can show that the resulting
g and g

are homotopic by building a homotopy from the homotopy H :

n+1
I X from
f to f

to a homotopy H
k+1
:
n+1
I X such that H
k+1
(
n+1
0
I) and that extends
the homotopies built over f and f

as in the preceding paragraphs. Then H

k+1
[
d
0

n+1
I
will
be a homotopy from g to g

. We leave the details to the reader.

Lemmas 9.5 and 9.6 together prove the following.
Proposition 9.7. If X is a Kan complex, the denitions of
n
(X, ) in Denitions 9.1 and
9.3 agree.
The group structure. One benet of the version of
n
(X, ) given in Denition 9.3,
compared to the perhaps more geometrically transparent Denition 9.1, is the ease of proving
that
n
(X, ) is a group and of describing the group operation.
Denition 9.8. Let x, y be two n-simplices, n 1, in the Kan complex X such that
d
i
x = d
i
y for all i. Let
n+1
n
be the horn of
n+1
in X such that the face corresponding
to d
n+1

n+1
is y, the face corresponding to d
n1

n+1
is x, and the faces corresponding to
all other sides of the horn are in . Let z be an extension of the horn to
n+1
as guaranteed
by the Kan condition. Then dene xy as the homotopy class of d
n
z in
n
(X, ). See Figure
29.
It can be shown that the denition is independent of the choices made:
Proposition 9.9. The product of Denition 9.8 yields a well-dened function
n
(X, )

n
(X, )
n
(X, ).
Proof. The proof is by various applications of the Kan extension condition. See [11, Lemma
4.2]. This would also be a good exercise for the reader.
The idea of the product on the simplicial
n
(X, ) is not far from that for the product
in the topological homotopy groups. First, suppose one has a map of the (n + 1)-ball D
n+1
to a topological space X such that the equator of the boundary sphere S
n
is mapped to
the basepoint of X. Then the restrictions of the map to the upper and lower hemispheres
of S
n
determine elements of
n
(X, ), and the map of all of D
n+1
determines a homotopy
between them. Secondly, recall that, roughly speaking, the product of two elements x, y in
42
Figure 29: The product of x and y in
1
(X, ) (above) or
2
(X, ) (below).
the topological
n
(X, ) can be represented by a map of a sphere that agrees with x and y
on two disjoint disks in S
n
and takes the rest of S
n
to the basepoint.
Denition 9.8 puts these ideas together. In the simplicial world, we can think of d
n

n+1
as being one hemisphere of
n+1
and the rest of
n+1
as the other hemisphere. Then in
Denition 9.8, the (n + 1)-simplex z can be thought of as providing a homotopy between
d
n
z and what is happening on the rest of z (notice that, indeed, d
n
z ). But the rest
of z contains x and y on two separate faces and everything else goes to , just as for the
topological product.
Of course we expect
n
(X, ) to be a group if n > 0.
Theorem 9.10. With the product of Denition 9.8,
n
(X, ) is a group.
Proof. The constructions are pictured in Figure 30.
The constant map
n
(which we will also denote by ) is the identity element.
Indeed, given x X representing an element of
n
(X, ), the (n + 1)-simplex s
n
x will have
d
n+1
s
n
x = d
n
s
n
x = x, while for i < n, d
i
s
n
x = s
n1
d
i
x . This realizes x = x. Similarly,
consideration of s
n1
x gives x = x.
It is also easy to construct inverses: given x X representing an element of
n
(X, ),
there is no problem mapping the horn
n+1
n+1
into X such that the face corresponding to
d
n1

n+1
goes to x and all other faces land in . The Kan condition lets us extend this to
a map of
n+1
into X and then the face corresponding to d
n+1

n+1
is a right inverse to x.
Similarly, we can nd a left inverse using
n+1
n1
and putting x on the face corresponding to
d
n+1

n+1
.
Finally, we show that the group operation is associative, which takes a bit more work.
Let x, y, z be simplices in X representing elements of
n
(X, ). We choose (n +1)-simplices
w
n1
and w
n+2
that respectively realize the products xy and yz, and we choose a simplex
w
n+1
realizing the product (xy)z, where xy is represented by d
n
w
n1
. Now, we can nd a
43
Figure 30: Above left: the identity x = x. Above right: construction of the right and left
inverses of x. Below: Associativity (xy)z = x(yz).
horn
n+2
n
in X such that the faces corresponding to d
i

n+2
are the w
i
for i = n 1, n +
1, n + 2 and otherwise. To see that this data is consistent to form the horn, we need
to check the appropriate faces, most of which are in , to see that they correspond. The
only faces of
n+2
we dont need to check are those of the form d
i
d
n

n+2
since d
n

n+2
isnt in the horn. By the simplicial axioms, these also correspond to the faces d
n1
d
i

n+2
for i < n and d
n
d
i+1

n+2
for i n. This leaves the following faces to check: We have
d
n
d
n1

n+2
n
= d
n
w
n1
= xy = d
n1
w
n+1
= d
n1
d
n+1

n+2
n
and d
n+1
d
n1

n+2
n
= d
n+1
w
n1
=
y = d
n1
w
n+2
= d
n1
d
n+2

n+2
n
. We also have d
n+1
d
n+1

n+2
n
= d
n+1
w
n+1
= z = d
n+1
w
n+2
=
d
n+1
d
n+2

n+2
n
. All other sides in the proposed horn are in , and so the data is consistent.
We can extend this horn to an (n + 2)-simplex u by the Kan condition. So now by
denition of w
n+1
, (xy)z = d
n
w
n+1
= d
n
d
n+1
u, which, using the axioms, is also equal to
d
n
d
n
u. But this also represents the product of d
n1
d
n
u = d
n1
d
n1
u = d
n1
w
n1
= x with
d
n+1
d
n
u = d
n
d
n+2
u = d
n
w
n+2
= yz. So d
n
d
n
u also represents the product x(yz), proving
associativity.
Also as expected,
n
(X, ) is an abelian group for n 2, but this is a bit more dicult
to prove. We refer the reader to [11, Proposition 4.4].
Relative homotopy groups. If (X, A, ) is a Kan triple (meaning A is a Kan subcomplex
of the Kan complex X and is a basepoint in A), there are also relative homotopy groups

n
(X, A, ). Corresponding to our rst denition of
n
(X, ) and the topological notion
of relative homotopy, we could dene
n
(X, A, ) to be relative homotopy classes of maps
(
n
,
n
, [0]) (X, A, ), where the homotopies are required to keep the image of
n
I
in A and the image of [0] I in . For a version of
n
(X, A, ) corresponding to our second
44
denition of
n
(X, ), we rst need a relative notion of homotopy of simplices:
Denition 9.11. If A is a subcomplex of X, we say that two n-simplices x, x

X
n
are
homotopic rel A if d
i
x = d
i
x

for 1 i n, d
0
x is homotopic to d
0
x

in A via an n-simplex
y, and there exists a simplex w X
n+1
such that d
0
w = y, d
n
w = x, d
n+1
w = x

, and
d
i
w = s
n1
d
i
x = s
n1
d
i
x

, 1 i n 1.
This denition is very similar to that for homotopy of simplices except instead of requiring
d
0
x = d
0
x

, we let d
0
x and d
0
x

be two simplices that are themselves homotopic in A, and

the homotopy between x and x

, provided by w, contains within it the homotopy between

d
0
x and d
0
x

.
Using this relative notion of homotopy, we can dene
n
(X, A, ).
Denition 9.12. Given a Kan triple (X, A, ), we dene
n
(X, A, ), n > 0, as the set of
equivalence classes of n-simplices x X with d
0
x A and d
i
x for all i, 1 i n, up
to relative homotopy of simplices.

n
(X, A, ) is also a group for n 2 and an abelian group for n 3. We will dene the
product; the proofs of well-denedness and that we have a group are analogous to those for

n
(X, ).
Denition 9.13. Suppose x, y represent elements of
n
(X, A, ), n 2. Let z represent
the product between d
0
x and d
0
y in
n1
(A, ). So z A is such that d
n2
z = d
0
x,
d
n
z = d
0
y and d
n1
z represents (d
0
x)(d
0
y). Now map the horn
n+1
n
into X such that the
sides corresponding to d
0

n+1
, d
n1

n+1
, and d
n+1

n+1
are z, x, and y, respectively, and all
other faces go to . One can check that this is consistent data. Then let w be an extension
of the horn, which exists because X is Kan, and dene xy to be d
n
w.
Figure 31: The product of two elements of x, y
2
(X, A, ). The 1-simplex with endpoints
1 and 3 represents the product (d
0
x)(d
0
y) in
1
(A).
An excellent exercise for the reader at this point would be to show that there is a long
exact sequence

n
(A, )
n
(X, )
n
(X, A, )
n1
(A, ) .
45
10 Concluding remarks
It is dicult to know where to end a survey of the type we have undertaken here. On the
one hand, although we have included some material from its later chapters, we have not even
covered the entire rst chapter of Mays textbook [11]! On the other hand, our goal has never
been to provide a completely rigorous or comprehensive treatise on simplicial theory, but
to provide the reader with an introduction to some of the most important elementary ideas
while maintaining a bridge to the geometric pictures that the combinatorics are based upon.
We hope that we have prepared the interested student to move on to the more standard
texts on simplicial objects with some picture (literally) of whats going on there.
And what is going on there? Just about everything in topological homotopy theory and
then some. Just a glance at the table of contents of [11] turns up many familiar concepts from
homotopy theory: brations, ber bundles, Postnikov systems, function spaces, Hurewicz
theorems, Eilenberg-Mac Lane complexes, k-invariants, cup and cap products, the Serre
spectral sequence, . . . . This is not surprising in light of the following theorem; we refer
the reader to Curtis [3, Section 12], or to [8, Section I.11] for a modern proof.
Theorem 10.1. The homotopy category of Kan complexes, consisting of Kan complexes and
homotopy classes of maps between them, is equivalent to the category of CW complexes and
homotopy classes of continuous maps.
The functors that realize this equivalence are the realization functor of simplicial com-
plexes and the singular set functor that assigns the singular set to a topological space. Thus
this theorem is closely related to the adjointness theorem, Theorem 4.10. So this tells us that
everything we have been doing in the simplicial realm is a reection of ordinary homotopy
theory. Yet, despite the geometric point of view we have been emphasizing here, simplicial
theory is purely combinatorial and algebraic, accessible by discrete tools that may not be
evident in pure topology. Thus, using simplicial theory, one can hope to study topological
homotopy theory via these combinatorial tools. Furthermore, we touched upon how the
combinatorial simplicial methods can be transported to other contexts, such as simplicial
groups. They can also be abstracted to broader categorical settings, leading to the theory
of model categories and simplicial categories. We hope to have introduced enough of the
background also to enable the reader to pursue these more modern approaches, such as can
be found in [8], with some understanding of their original motivation in concrete homotopy
theory.
We leave the reader with some bibliographical notes on the sources we have used.
Our primary sources were Mays Simplicial Objects in Algebraic Topology [11] and Moores
lecture notes Seminar on algebraic homotopy theory [13]. Mays book, rst published in 1967,
is the most comprehensive reference of its time, featuring a direct combinatorial approach.
Moores notes are from nearly a decade earlier, but they are perhaps a bit more accessible to
the geometrically-minded reader; they take a dierent approach to homotopy groups, dening
them as
0
of simplicial loops spaces. Our primary modern source was Simplicial Homotopy
Theory [8] by Goerss and Jardine. It starts o directly from the modern model category
point of view, without much need for the combinatorial underpinnings (some knowledge of
46
the combinatorial approach, however, will aid the reader). Despite the abstractness of the
material, I found this book quite readable. The book Calculus of Fractions and Homotopy
Theory [7] by Gabriel and Zisman, though contemporary with Mays book, is something of
a bridge between the classical combinatorics and some of the more current axiomatic ideas.
We should also mention in this paragraph the long survey Simplicial Homotopy Theory [3]
by Curtis. As one might expect, each of these sources contains somewhat dierent material
and sometimes dierent approaches to the same material, thus it is well worth consulting
each of them depending on the readers interests in terms of both material and style.
Besides these longer expositions, short introductory chapters on simplicial theory can
be found within many other textbooks and surveys. In particular, I know of sections on
simplicial theory in Selicks Introduction to Homotopy Theory [19], Smirnovs Simplicial and
Operad Methods in Algebraic Topology [20], and Weibels An Introduction to Homological
Algebra [21]. As one might expect, this last reference is a good source for applications of
simplicial theory to homological algebra. There are also review sections on simplicial sets
in Bouseld and Kans Homotopy Limits, Completions, and Localizations [1] and in Mixed
Hodge Structures [15] by Peters and Steenbrink. The breadth of topics covered by those
titles alone should give the reader some impression of just how varied the applications of
simplicial theory are.
References
[1] A. K. Bouseld and D. M. Kan, Homotopy limits, completions and localizations, Lecture
Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin, 1972.
[2] Glen Bredon, Topology and geometry, Springer-Verlag, New York, 1993.
[3] Edward B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107209.
[4] James F. Davis and Paul Kirk, Lecture notes in algebraic topology, Graduate Studies in
Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001.
[5] Samuel Eilenberg and J. A. Zilber, Semi-simplicial complexes and singular homology,
Ann. of Math. (2) 51 (1950), 499513.
[6] Yves Felix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory,
Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001.
[7] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der
Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New
York, 1967.
[8] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathe-
matics, vol. 174, Birkhauser Verlag, Basel, 1999.
[9] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
47
[10] J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin, 1972, Lectures
Notes in Mathematics, Vol. 271.
[11] J. Peter May, Simplicial objects in algebraic topology, University of Chicago Press,
Chicago, IL, 1992.
[12] John Milnor, The geometric realization of a semi-simplicial complex, Ann. of Math. (2)
65 (1957), 357362.
[13] J.C. Moore, Seminar on algebraic homotopy theory, Mimeographed notes - Princeton,
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[14] James R. Munkres, Elements of algebraic topology, Addison-Wesley, Reading, MA, 1984.
[15] Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, Ergebnisse
der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Math-
ematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern
Surveys in Mathematics], vol. 52, Springer-Verlag, Berlin, 2008.
[16] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205295.
[17] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43,
Springer-Verlag, Berlin, 1967.
[18] C. P. Rourke and B. J. Sanderson, -sets. I. Homotopy theory, Quart. J. Math. Oxford
Ser. (2) 22 (1971), 321338.
[19] Paul Selick, Introduction to homotopy theory, Fields Institute Monographs, vol. 9, Amer-
ican Mathematical Society, Providence, RI, 1997.
[20] V. A. Smirnov, Simplicial and operad methods in algebraic topology, Translations of
Mathematical Monographs, vol. 198, American Mathematical Society, Providence, RI,
2001, Translated from the Russian manuscript by G. L. Rybnikov.
[21] Charles A. Weibel, An introduction to homological algebra, Cambridge studies in ad-
vanced mathematics, Cambridge University Press, Cambridge, 1994.
Some diagrams in this paper were typeset using the T
E
X commutative diagrams package
by Paul Taylor.
48