Concrete categories and monomorphisms

This post serves to collect some background on concrete categories for my next post.

Concrete categories are categories in which objects have an underlying set:

Definition. A concrete category is a pair $(\mathscr C, U)$ of a category $\mathscr C$ with a faithful functor $U \colon \mathscr C \to \mathbf{Set}$ . In cases where $U$ is understood, we will simply say $\mathscr C$ is a concrete category.

Example. The categories $\mathbf{Gp}$ of groups, $\mathbf{Top}$ of topological spaces, $\mathbf{Ring}$ of rings, and $\mathbf{Mod}_R$ of $R$ -modules are concrete in an obvious way. The category $\mathbf{Sh}(X)$ of sheaves on a site $X$ with enough points is concrete by mapping a sheaf to the disjoint union of its stalks (the same holds for any Grothendieck topos, but a different argument is needed). Similarly, the category $\mathbf{Sch}$ of schemes can be concretised by sending $(X,\mathcal O_X)$ to $\coprod_{x \in X} \mathcal P(\mathcal O_{X,x})$ , where $\mathcal P$ is the contravariant power set functor.

Today we will study the relationship between monomorphisms and injections in $\mathscr C$ :

Lemma. Let $(\mathscr C,U)$ be a concrete category, and let $f \colon A \to B$ be a morphism in $\mathscr C$ . If $Uf$ is a monomorphism (resp. epimorphism), then so is $f$ .

Proof. A morphism $f \colon A \to B$ in $\mathscr C$ is a monomorphism if and only if the induced map $\Mor_{\mathscr C}(-,A) \to \Mor_{\mathscr C}(-,B)$ is injective. Faithfulness implies that the vertical maps in the commutative diagram

$\begin{array}{ccc} \Mor_{\mathscr C}(-,A) & \to & \Mor_{\mathscr C}(-,B) \\ \downarrow & & \downarrow \\ \Mor_{\mathbf{Set}}(U-,UA) & \to & \Mor_{\mathbf{Set}}(U-,UB) \end{array}$

are injective, hence if the bottom map is injective so is the top. The statement about epimorphisms follows dually. $\qedsymbol$

For example, this says that any injection of groups is a monomorphism, and any surjection of rings is an epimorphism, since the monomorphisms (epimorphisms) in $\mathbf{Set}$ are exactly the injections (surjections).

In some concrete categories, these are the only monomorphisms and epimorphisms. For example:

Lemma. Let $(\mathscr C,U)$ be a concrete category such that the forgetful functor $U$ admits a left (right) adjoint. Then every monomorphism (epimorphism) in $\mathscr C$ is injective (surjective).

Proof. If $U$ is a right adjoint, it preserves limits. But $f \colon A \to B$ is a monomorphism if and only if the square

$\begin{array}{ccc} A & \overset{\text{id}}\to & A \\ \!\!\!\!\!{\scriptsize \text{id}}\downarrow & & \downarrow {\scriptsize f}\!\!\!\!\! \\ A & \underset{f}\to & B \end{array}$

is a pullback. Thus, $U$ preserves monomorphisms if it preserves limits. The statement about epimorphisms is dual. $\qedsymbol$

For example, the forgetful functors on algebraic categories like $\mathbf{Gp}$ , $\mathbf{Ring}$ , and $\mathbf{Mod}_R$ have left adjoints (a free functor), so all monomorphisms are injective.

The forgetful functor $\mathbf{Top} \to \mathbf{Set}$ has adjoints on both sides: the left adjoint is given by the discrete topology, and the right adjoint by the indiscrete topology. Thus, monomorphisms and epimorphisms in $\mathbf{Top}$ are exactly injections and surjections, respectively.

On the other hand, in the category $\mathbf{Haus}$ of Hausdorff topological spaces, the inclusion $\mathbf Q \hookrightarrow \mathbf R$ is an epimorphism that is not surjective. Indeed, a map $f \colon \mathbf R \to X$ to a Hausdorff space $X$ is determined by its values on $\mathbf Q$ .

Lovely little lemmas

Or maybe ‘amusing observations’

Concrete categories and monomorphisms

Leave a Reply Cancel reply