Pullback and Pushout

Subobjects

Subobject

A subobject of an object $X$ in a category $\mathbf{C}$ is a monomorphism $m:M\rightarrowtail X$Given subobjects $m,m^{\prime }$ of $X$, a morphism $f:m\to m^{\prime }$ is an morphism in $\mathbf{C}/X$. Thus we have a category ${\mathrm{Sub}}_{\mathbf{C}}(X)$ of subobjects of $X$ in $\mathbf{C}$.

Inclusion of Subobjects

We define the relation of inclusion of subobjects by: $m\subseteq m^{\prime }⟺\exists f:m\to m^{\prime }\in \text{mor}(\mathbf{C}/X)⟺\exists \varphi :M\to M^{\prime }.m=m^{\prime }\circ \varphi$

Equivalence of Subobjects

We say that subobjects $m$ and $m^{\prime }$ are equivalent if $m\subseteq m^{\prime }$ and $m^{\prime }\subseteq m$, write $m\equiv m^{\prime }$.

Proposition

Equivalent subobjects have isomorphic domains.

Proof Observe that, if $m\equiv m^{\prime }$ then $m=m^{\prime }f=m{f}^{\prime }f$, and since $m$ is monic, $f^{\prime }f=1_M$ and similarly $f{f}^{\prime }=1_{M^{\prime }}$. So $M\cong M^{\prime }$ via $f$. $\square$

Remark

We sometimes abuse notation and language by calling $M$ the subobject when the monomorphism $m:M\to X$ is clear.

Local Membership

In terms of generalized elements $z:Z\to X$ of an object $X$, we define the local membership relation as $z{\in }_{X}M$

Pullback

Pullback

In any category $\mathsf{C}$, a pullback of morphisms $f,g$ with $\mathrm{cod}(f)=\mathrm{cod}(g)$ consists of an object $P$ together with morphisms $p_1$, $p_2$ such that $f\circ p_1=g\circ p_2$, and universal with this property. i.e., given any $z_1:Z\to A$ and $z_2:Z\to B$ with $f\circ z_1=g\circ z_2$, there exists a unique $u:Z\to P$ with $z_1=p_1\circ u$ and $z_2=p_2\circ u$:

We write such $P$ as $A{\times }_{X}B$.

e.g. In $\mathsf{Set}$, the pullback of functions $f:X\to Z$ and $g:Y\to Z$ always exists and is given by $X{\times }_{Z}Y=\{(x,y)\in X\times Y\mid f(x)=g(y)\}.$

Corollary

If a category $\mathsf{C}$ has binary products and equalizers, then it has pullbacks.

e.g. An explicit construction of a pullback in $\mathsf{Set}$ of objects $f:A\to C$ and $g:B\to C$ as a subset of the product: $\{⟨a,b⟩\mid fa=gb\}=A{\times }_{C}B\hookrightarrow A\times B$

Proposition

Given a pullback $A{\times }_{X}B$ in any category:

If $g$ is monic, then $p_1$ is monic as well.

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 $E{\times }_A(A{\times }_{C}B)\cong E{\times }_{C}B$
• 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 $h:C^{\prime }\to C$ in a category $\mathbf{C}$ with pullbacks, there is a functor$h^{\ast }:\mathbf{C}/C\to \mathbf{C}/C^{\prime }$defined by $(\alpha :A\to C)\mapsto ({\alpha }^{\prime }:C^{\prime }{\times }_{C}A\to C^{\prime })$. We call ${\alpha }^{\prime }$ the pullback of $\alpha$ along $h$.
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 $\mathbf{C}$ has pullbacks and a terminal object $1$.

Pushout

Pushout

In any category $\mathsf{C}$, a pushout of morphisms $f,g$ with $\mathrm{dom}(f)=\mathrm{dom}(g)$ consists of an object $P$ together with morphisms $p_1$, $p_2$ such that $p_1\circ f=p_2\circ g$, and universal with this property. i.e., given any $z_1:Z\to A$ and $z_2:Z\to B$ with $z_1\circ f=z_2\circ g$, there exists a unique $u:Z\to P$ with $z_1=p_1\circ u$ and $z_2=p_2\circ u$:

We write such $P$ as $A{\bigsqcup }_{X}B$.