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.

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