nLab
poset of subobjects

Context

Category theory

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

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

Given any object XX in any category CC, the subobjects of XX form a poset, called (naturally enough) the poset of subobjects of XX, or the subobject poset of XX.

Sometimes it is the poset of regular subobjects that really matters (although these are the same in any (pre)topos).

Properties

If CC is finitely complete, then the subobjects form a meet-semilattice, so we may speak of the semilattice of subobjects.

In any coherent category (such as a pretopos), the subobjects form a distributive lattice, so we may speak of the lattice of subobjects.

In any Heyting category (such as a topos), the subobjects of XX form a Heyting algebra, so we may speak of the algebra of subobjects.

The reader can probably think of other variations on this theme.

Revised on March 15, 2012 15:47:35 by Urs Schreiber (82.172.178.200)