Subobjects and Pullback

Subobjects

Subobject

A subobject of an object in a category is a monomorphism Given subobjects of , a morphism is an morphism in . Thus we have a category of subobjects of in .

Inclusion of Subobjects

We define the relation of inclusion of subobjects by:

Equivalence of Subobjects

We say that subobjects and are equivalent if and , write .

Proposition

Equivalent subobjects have isomorphic domains.

Proof Observe that, if then , and since is monic, and similarly . So via .

Comment

We sometimes abuse notation and language by calling the subobject when the monomorphism is clear.

Definition

Def Local Membership In terms of generalized elements of an object , we define the local membership relation as

Pullback

Pullback

In any category , a pullback of morphisms with consists of morphisms such that , and universal with this property. i.e., given any and with , there exists a unique with and : And we write such as .

Corollary

If a category has binary products and equalizers, then it has pullbacks.

e.g. An explicit construction of a pullback in of objects and as a subset of the product:

Proposition

Prop Given a pullback in any category: If is monic, then is monic.

Lemma

Lemma Two-Pullbacks Consider the commutative diagram below in a category with pullbacks:

• If the two squares are pullbacks, so is the outer rectangle. That is
• If the right square and outer square are pullbacks, so is the left square.

Corollary The pullback of a commutative triangle is a commutative triangle. Specifically, given a commutative triangle as on the right end of the following “prism diagram”:

Prop For fixed in a category with pullbacks, there is a functordefined by . We call the pullback of along . Proof

Prop A category has finite products and equalizers iff it has pullbacks and a terminal object. Proof The “only if” direction has already been done. For the other direction, suppose has pullbacks and a terminal object .