nLab coframe

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

• a poset

• that has

  • all small products, called meets $\bigwedge$

  • all finite colimits, called joins $\vee$

• 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.

See also

• co-Heyting algebra

• frame

• cotopos

 References

• Mamuka Jibladze, “Cosheaves, coframes, cotoposes: some new facts, some old questions” (web archive)