subobject

Let $\mathcal{C}$ be a category.

Equivalent Monomorphisms and Subobjects

Let $\operatorname{Mono}(-,B)$ be the class of all monomorphisms into object $B$ in $\mathcal{C}$ . Two elements $f\colon A_1\to B$ and $g\colon A_2\to B$ in $\operatorname{Mono}(-,B)$ are equivalent if there exist two morphisms $r:A_1\rightarrow A_2$ and $s:A_2\rightarrow A_1$ such that we have the following two commutative diagrams:

It is easy to see that both $r$ and $s$ are monomorphic. In fact, $A_1$ and $A_2$ are isomorphic objects, as $f=g\circ r$ and $g=f\circ s$ , so that $f=(f\circ s)\circ r$ , or $1_{A_1}=s\circ r$ , since $s\circ r$ is monomorphic. Similarly $1_{A_2}=r\circ s$ . Monomorphism equivalence is an equivalence relation on $\operatorname{Mono}(-,B)$ .

Definition. A subobject of an object $X$ in $\mathcal{C}$ is an equivalence class in $\operatorname{Mono}(-,X)$ . Let's write $$[A\to X]$$ for a subobject of $X$ , and a monomorphism $f:A\to X$ a representative of $[A\to X]$ . If there is no danger of confusion, it is often easier to identify a subobject $[A\to X]$ by $A$ , and simply write $$A\subseteq X,$$ as long as we keep in mind that, along with the object $A$ , there is a monomorphism from $A$ to $X$ . The class of subobjects of $X$ shall be denoted by $$\Sub(X):=\lbrace A\mid A\subseteq X\rbrace.$$ Definition. A category $\mathcal{C}$ is said to be well-powered or locally small if for every object $X$ in $\mathcal{C}$ , the class $\Sub(X)$ is a set. Most common categories are locally small.

Suppose now that $\mathcal{C}$ is well-powered and has pullbacks (a pullback exists for every pair of morphisms into the same object). We shall turn $\Sub$ into a functor from $\mathcal{C}$ to Set. For every morphism $\alpha: X\to Y$ , define $\Sub(\alpha)$ as follows:

take a representative $g\in[B\to Y]$ , consider the pullback of $\alpha$ and $g$ indicated in the commutative diagram below:

Since $g$ is monomorphism, so is $f:A\to X$ , and hence $A$ is a subobject of $X$ . We set $\Sub(\alpha)([g]):=[f]$ .

Because the diagram is a pullback, $\Sub(\alpha)([g])$ gives a unique value, and thus $$\Sub(\alpha):\Sub(Y)\to\Sub(X)$$ is a well-defined morphism. Furthermore, it is easily verified that $\Sub(1_X)=1_{\Sub(X)}$ and $\Sub(\alpha\circ\beta)= \Sub(\beta)\circ\Sub(\alpha)$ . Thus, $\Sub$ is a contravariant functor from $\mathcal{C}$ to ${Set}$ (or a covariant functor $\mathcal{C}^{op}\to {Set}$ ), and is called the subobject functor of $\mathcal{C}$ .

Equivalent Epimorphisms and Quotient Objects

Dually, given an object $A$ in a category $\mathcal{C}$ , we can define an equivalence relation on $\operatorname{Epi}(A,-)$ , the class of all epimorphisms from $A$ , by reversing all arrows in the previous paragraph. Specifically, two elements $f\colon A \to B_1$ and $g\colon A \to B_2$ in $\operatorname{Epi}(A,-)$ are equivalent if there exist two morphisms $B_1\rightarrow B_2$ and $B_2\rightarrow B_1$ such that the following two diagrams commute:

Definition. A quotient object of $X$ is an equivalence class in $\operatorname{Epi}(X,-)$ . A typical quotient object is denoted by $\lbrack X\to B \rbrack$ . If $\mathcal{C}$ is a small category and has pushouts, then there is a covariant functor $\Quo:\mathcal{C}\to {Set}$ taking each object of $\mathcal{C}$ to its set of quotient objects and each morphism between two objects to a morphism between the sets of their quotient objects. $\Quo$ is called the quotient object functor of $\mathcal{C}$ .