This entry is about the notion of frame in topos theory. For other notions, see frame (disambiguation).
The notion of frame is a generalization of the notion of category of open subsets of a topological space. A frame is like a category of open subsets in a space possibly more general than a topological space: a locale. This in turn is effectively defined to be anything that has a collection of open subsets that behaves essentially like the open subsets of a topological space do.
A frame is
a poset
that has
all small coproducts, called joins
all finite limits, called meets
and which satisfies the infinite distributive law:
for all in
(Note that the converse holds in any case, so we have equality.)
A frame homomorphism is a homomorphism of posets that preserves finite meets and arbitrary joins. Frames and frame homomorphisms form the category Frm.
The formal duals of frames, hence the objects in the opposite category Locale Frm are called locales.
The notation to supposed to remind us that every frame is like a frame of opens of the corresponding locale.
So in particular a frame is a lattice which has not only finite joins but all small joins. Being small itself, it is a suplattice, and hence a complete lattice: it also has all (small) meets.
Furthermore, as the distributive law certainly holds when the joins in question are finite, it is a distributive lattice.
A useful way to understand frames and locales is as the simplest nontrivial special cases of (n,1)-toposes. So we start in
with some remarks on this, and only then turn to
of frames, which should make more sense this way.
The notion of frame – or rather its formal dual, the notion of locale – is the special case of the notion of (n,1)-toposes for : (0,1)-toposes.
The axioms on a frame are nothing but Giraud's axioms on sheaf toposes, restricted to (0,1)-categories:
given the existence of finite limits and arbitrary colimits, the infinite distributive law expresses that a frame has universal colimits: they are stable under pullback. (For notice that in a poset pullbacks and products coincide.)
Then a morphism of frames is precisely (the inverse image of) a geometric morphism: a morphism preserving finite limits and arbitrary colimits.
A frame is automatically a Heyting algebra.
In category theoretic terms this means that it ia a cartesian closed category, hence that
for every object the functor
that forms the product with has a right adjoint
This exists by the adjoint functor theorem, using that there is only a finite number of morphisms between any two objects (one or none) and that finite limits exist in .
Therefore one may think of a of a frame as a cartesian closed suplattice, or a cartesian quantale.
But notice that the frame homomorphisms are not required to preserve the Heyting implication.
A frame is naturally equipped with the structure of a site:
a family of morphism is covering precisely if is the union of the
For more on this see locale.
The opposite category to the category Frm is the category Loc of locales (possibly slightly generalized topological spaces)
Conversely, any topological space has a frame of open subsets. (In fact, one definition of a topological space is a set equipped with a subframe of its power set.)
For a topological space, the category of open subsets of is a frame: the frame of opens.
For a set, the power set of is the frame of opens corresponding to the discrete topology on .
The poset of truth values is a frame equivalent to the frame of opens corresponding to the discrete topology on a singleton.
For a geometric category and an object in , the subobject poset is a frame.
A complete decidable linear order is a frame.
For any and collection , the inequality holds automatically. For the reverse inequality, we note this follows trivially in case , since in that case we have for all , whence
Otherwise we are in the case , where we must show the inequality
But this inequality must hold, else which would imply for all , whence , contradiction.
Frames in univalent foundations:
Last revised on September 13, 2024 at 22:32:33. See the history of this page for a list of all contributions to it.