# nLab poset of subobjects

category theory

(0,1)-category

(0,1)-topos

## Theorems

#### Topos Theory

Could not include topos theory - contents

# Contents

## Idea

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

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

## Properties

If $C$ 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 $X$ 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)