Could not include topos theory - contents
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
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.
Furthermore, as the distributive law certainly holds when the joins in question are finite, it is a distributive lattice.
with some remarks on this, and only then turn to
of frames, which should make more sense this way.
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.)
A frame is automatically a Heyting algebra.
for every object the functor
This exists by the adjoint functor theorem, using that there is only a finite number of morphims between any two objects (one or none) and that finite limits exist in .
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:
For more on this see locale.