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 (though in general it may be a large one, i.e. a partially ordered proper class; cf. well-powered category). This is 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.

  • If one opts for the alternative1 definition that subobjects are monomorphisms into the object (not isomorphism classes thereof), then one gets a preorder of subobjects instead. In any case, the poset of subobjects Sub(X)Sub(X) in our sense is the posetal reflection of the preorder Mono(X)Mono(X) of subobjects in the alternative sense, and of course the reflection quotient map Mono(X)Sub(X)Mono(X) \to Sub(X) is an equivalence.

References


  1. Discussions of this can be found in A.1.3 of Johnstone’s Elephant, and also this MO discussion.

Revised on July 15, 2017 08:55:34 by Todd Trimble (24.146.226.222)