Contents

topos theory

(0,1)-category

(0,1)-topos

# Contents

## Idea

The notion of coframe is a generalization of the notion of category of closed subsets? of a topological space. A coframe is like a category of closed 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 closed subsets that behaves essentially like the closed subsets of a topological space do.

It is also the opposite poset of a frame.

## Definition

###### Definition

A coframe $\mathcal{C}$ is

• that has

• and which satisfies the infinite distributive law:

$\bigwedge_i (x\vee y_i) \leq x \vee (\bigwedge_i y_i)$

for all $x, \{y_i\}_i$ in $A$

(Note that the converse holds in any case, so we have equality.)

A coframe homomorphism is a homomorphism of posets that preserves finite joins and arbitrary meets. Coframes and coframe homomorphisms form the category Cofrm.