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Idea

A simplicial object $X$ in a category $C$ is a simplicial set internal to $C$: a collection $\{X_n\}_{n \in \mathbb{N}}$ of objects in $C$ that behave as if $X_n$ were an object of $n$-dimensional simplices internal to $C$ equipped with maps between these spaces that assign faces and degenerate simplices.

For instance, and there is a longer list further down this page, a simplicial object in $Grps$ is a collection $\{G_n\}_{n\in \mathbb{N}}$ of groups, together with face and degeneracy homomorphisms between them. This is just a simplicial group. We equally well have other important instances of the same idea, when we replace $Grps$ by other categories, or higher categories.

Definition

A simplicial object in a category $C$ is a functor $\Delta^{op} \to C$, where $\Delta$ is the simplicial indexing category.

More generally, a simplicial object in an (∞,1)-category is an (∞,1)-functor $\Delta^{op} \to C$.

A cosimplicial object in $C$ is similarly a functor out of the opposite category, $\Delta \to C$.

Accordingly, simplicial and cosimplicial objects in $C$ themselves form a category in an obvious way, namely the functor category $[\Delta^{op},C]$ and $[\Delta,C]$, respectively.

Remark

A simplicial object $X$ in $C$ is often specified by the objects, $X_n$, which are the images under $X$, of the objects $[n]$ of $\Delta$, together with a description of the face and degeneracy morphisms, $d_i$ and $s_j$, which must satisfy the simplicial identities.

Examples

• A simplicial object in Sets is a simplicial set.

• A simplicial object in Presheaves is a simplicial presheaf.

• A simplicial object in TopologicalSpaces is a simplicial topological space.

• A simplicial object in Manifolds is a simplicial manifold.

• A simplicial object in Groups is a simplicial group.

• A simplicial object in AbelianGroups is a simplicial abelian group.

• A simplicial object in TopologicalGroups is a simplicial topological group.

• A simplicial object in Lie algebras is a simplicial Lie algebra.

• A simplicial object in Rings is a simplicial ring.

• A cosimplicial object in the category of rings (algebras) is a cosimplicial ring (cosimplicial algebra).

• A simplicial object in a category of simplicial objects is a bisimplicial object.

• A cosimplicial object in sSet is a cosimplicial simplicial set (equivalently a simplicial object in cosimplicial sets).

• The bar construction produces a simplicial object from a monad and an algebra over that monad.

Category of simplicial objects

For $D$ a category, we write $D^{\Delta^{op}}$ for the functor category from $\Delta^{op}$ to $D$: its category of simplicial objects.

Simplicial enrichment

Geometric realization

If $D$ is already a simplicially enriched category in its own right, with powers and copowers, we can define the geometric realization of a simplicial object $X\in D^{\Delta^{op}}$ as a coend:

where $\odot$ denotes the copower for the simplicial enrichment of $D$. This is left adjoint to the “total singular object” functor $D \to D^{\Delta^{op}}$ sending $Y$ to the simplicial object $n\mapsto Y^{\Delta[n]}$, the power for the simplicial enrichment.

Perhaps surprisingly, this adjunction is even a simplicially enriched adjunction when $D^{\Delta^{op}}$ has its simplicial structure from Definition 1, even though the latter makes no reference to the given simplicial enrichment of $D$. A proof can be found in RSS01, Proposition 5.4.

Related concepts

• Reedy model structure

• simplex, simplex category

• simplicial object

  • simplicial set

  • simplicial diagram

  • simplicial object in an (∞,1)-category

• semi-simplicial object

  • semisimplicial set

• globular set, cubical set

• nerve, nerve and realization

• Segal condition

References

The original definition of simplicial objects, maps between them, and homotopies of such maps is due to Daniel M. Kan:

• Daniel M. Kan, On the homotopy relation for c.s.s. maps, Boletín de la Sociedad Matemática Mexicana 2 (1957), 75–81. PDF.

An early discussion of simplicial objects is in:

• Alexander Grothendieck, p. 108 (11 of 21) in: FGA Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf, English translation: pdf web version, web version).

See also:

• Peter May, Simplicial objects in algebraic topology, University of Chicago Press (1967) [[ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), djvu, pdf]

• Dai Tamaki, Akira Kono, Appendix A.1 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Charles Rezk, Stefan Schwede, and Brooke Shipley, Simplicial structures on model categories and functors, arxiv

• Michael Barr, John Kennison, Robert Raphael, Contractible simplicial objects, Comm. Math. Univ. Carol. 60 4 (2019) 473–495 [[pdf](https://math.mcgill.ca/barr/papers/cso.pdf) eudml:295068]