nLab coframe

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

(0,1)(0,1)-Category theory

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 𝒞\mathcal{C} is

  • a poset

  • that has

  • and which satisfies the infinite distributive law:

    i(xy i)x( iy i) \bigwedge_i (x\vee y_i) \leq x \vee (\bigwedge_i y_i)

    for all x,{y i} ix, \{y_i\}_i in AA

    (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

 References

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

Last revised on June 3, 2022 at 10:08:15. See the history of this page for a list of all contributions to it.