Categories and functors

1. The initial or the empty category has no objects, and hence no morphisms. The terminal category has only one object and its identity morphism.

2. \(\Set\), the category of sets: \(\Ob \ \Set\) is the collection of sets, \(\Mor(X, Y)\) is the set of functions \(X \rightarrow Y\), and composition is the usual composition of functions.

   We will refer to the category \(\Set\) as the ‘category of sets and functions (between sets)’, since the composition is understood, or even as just ‘the category of sets’, when both morphisms and their composition rules are understood.

   Terminological/notational note. In what follows, we will usually omit describing the composition, and sometimes the morphisms too, but in each case what we omit will be understood from the context.

3. \(\Grp\), the category of groups and group homomorphisms.

4. The category, say \(\overline{\Grp}\), whose objects are the groups, and where \(\Mor(G, H)\) is the set of equivalence classes of group homomorphisms \(G \rightarrow H\), where \(f_1 \sim f_2\) if there exists \(h \in H\) such that \(f_1 = \Int h \circ f_2\), where \(\Int h =\) conjugation by \(h\). Composition is induced by the usual composition of homomorphisms: check that it is well-defined.

5. \(\Ab\), abelian groups and group homomorphisms.

6. \(\Top\), topological spaces and continuous maps.

7. \(\HTop\), topological spaces and homotopy classes of continuous maps between them: the well-definedness of composition involves checking, e.g., that if \(f_1, f_2 : X \rightarrow Y\) are homotopic to each other and \(g_1, g_2 : Y \rightarrow Z\) are homotopic to each other, then \(g_1 \circ f_1, g_2 \circ f_2 : X \rightarrow Z\) are homotopic to each other.

8. \(\Man\), manifolds and smooth maps.

9. \(\Ring\), rings and ring homomorphisms. In this course, rings are required to contain a multiplicative identity (unless otherwise stated), and the multiplication in a ring is required to be associative, though not necessarily commutative.

   Sometimes, we may consider the category \(\Rng\) of “rings without identity”, whose objects satisfy all that is required of a ring except that they may or may not have a multiplicative identity (and hence homomorphisms are not required to preserve a multiplicative identity either).

10. For a commutative ring \(R\), recall that an \(R\)-algebra is a ring \(S\) together with a ring homomorphism \(\iota : R \rightarrow S\), called the structure morphism, such that \(\iota(R)\) is contained in the center \(Z(S)\) of \(S\) (i.e., \(\iota(R)\) commutes with every element of \(S\)). Then we have the category of \(R\)-algebras: its objects are \(R\)-algebras, and the morphisms between two objects \((S_1, \iota_1)\) and \((S_2, \iota_2)\) are ring homomorphisms \(f : S_1 \rightarrow S_2\) fitting into a commutative diagram

    

    In practice, \(\iota\) will be suppressed from notation: when we say that \(S\) is an \(R\)-algebra, it will be understood that a structure morphism \(R \rightarrow S\) has been given, and whenever we write \(R \rightarrow S\) without further comment, it will be understood to refer to the structure morphism.

11. We also have the following category \(\Alg_R\) of commutative \(R\)-algebras: these are \(R\)-algebras \((S, \iota)\) such that the ring \(S\) is commutative.²

12. For a not necessarily commutative ring \(R\), we have the category \({}_R\Mod\) (resp., \(\Mod_R\)) of left \(R\)-modules (resp., right \(R\)-modules) and \(R\)-module homomorphisms. Note that \({}_{\ZZ}\Mod\) can be identified with \(\Ab\).

13. \(\Vect_k := {}_k\Mod\), when \(k\) is a field, so this is the category of vector spaces over \(k\) and \(k\)-linear transformations.

14. \(\Ban_{\RR}\) (resp., \(\Ban_{\CC}\)), Banach spaces over \(\RR\) (resp., \(\CC\)) and bounded linear maps.

15. Given a group \(G\), the category of \(G\)-sets, i.e., sets \(X\) together with an action of \(G\), where \(\Mor(X, Y)\) is the set of maps \(X \rightarrow Y\) respecting the \(G\)-actions.

16. The category of pairs \((G, X)\) where \(G\) is a group acting on a set \(X\); \(\Mor((G, X), (H, Y))\) consists of all pairs consisting of a homomorphism \(G \rightarrow H\) and a function \(X \rightarrow Y\) with the obvious compatibility: if the former maps \(g\) to \(h\) and the latter \(x\) to \(y\), the latter also maps \(g \cdot x\) to \(h \cdot y\).

17. Pairs \((V, T)\) consisting of a vector space \(V\) over a given field \(k\), and a \(k\)-linear transformation \(T : V \rightarrow V\), with \[\Mor((V, T), (W, U)) = \{f : V \rightarrow W \mid f \circ T = U \circ f\}.\]

18. Open subsets of \(\mathbb{C}^n\) and holomorphic maps between them.

19. Given a commutative ring \(R\) and a group \(G\), the category whose objects are \(R\)-modules equipped with a \(G\)-action by \(R\)-module automorphisms, and whose morphisms are morphisms of \(R\)-modules that respect the \(G\)-action. We will denote this category by \(\Rep_R(G)\). If \(R = k\) is a field, this is by definition the category of representations of \(G\) on \(k\)-vector spaces.

Let \(\{\mcC_i \mid i \in I\}\) be a family of categories indexed by a set \(I\).

• We then have a product category, \(\prod_{i \in I} \mcC_i\), as follows:

  • \(\Ob \left( \prod_{i \in I} \mcC_i \right) := \prod_{i \in I} \Ob \mcC_i\).

  • \(\Mor_{\prod_{i \in I} \mcC_i} ((X_i)_i, (Y_i)_i) = \prod_{i \in I} \Mor_{\mcC_i}(X_i, Y_i)\),

  with component-wise composition. If \(I = \emptyset\), then \(\prod_i \mcC_i\) is the terminal category (see Example ↗(i)).

• We also have a coproduct or a “disjoint union” category, \(\coprod_{i \in I} \mcC_i\):

  • \(\Ob \left( \coprod_{i \in I} \mcC_i \right) := \coprod_{i \in I} \Ob \mcC_i\).

  • \(\Mor_{\coprod_{i \in I} \mcC_i}(X_j, Y_k) := \begin{cases} \Mor_{\mcC_j}(X_j, Y_k), & \text{if $j = k$, and} \\ \emptyset, & \text{if $j \neq k$.} \end{cases}\)

  If \(I = \emptyset\), then \(\coprod_i \mcC_i\) is the empty or the initial category (see Example ↗(i)).

If \(G\) is a group, define \(*_G\) to be the category such that \(\Ob *_G = \{*\}\) is a singleton set, \(\Mor(*, *) = G\), and composition of morphisms is multiplication in \(G\).

1. Here are two full subcategories of \(\Alg_R\):

   • the category \(\Alg_R^{\ft}\) of finite type \(R\)-algebras: by definition, these are the \(R\)-algebras that are finitely generated over \(R\), i.e., they are isomorphic to \(R\)-algebras of the form \(R[x_1, \dots, x_n]/I\), where \(R[x_1, \dots, x_n]\) is the polynomial ring in \(n\) variables over \(R\), which is an \(R\)-algebra in an obvious way, and \(I \subset R[x_1, \dots, x_n]\) is some ideal.

   • the category \(\Alg_R^{\fin}\) of finite \(R\)-algebras: these are \(R\)-algebras \(S\) with the property that \(S\), when viewed as an \(R\)-module (using the structure morphism \(R \rightarrow S\)), is finitely generated.

2. \({}_R\Mod\) has a full subcategory \({}_R\Mod^{\fg}\) of finitely generated \(R\)-modules, and similarly we have \(\Mod_R^{\fg}\). Similarly, \(\Vect_k\) has a full subcategory \(\Vect_k^{\fd} := {}_k\Mod^{\fg}\) of finite dimensional vector spaces.

In familiar categories, the categorical notion of isomorphism recovers the usual one:

• a bijection in \(\Set\);

• an isomorphism of groups in \(\Grp\), of rings in \(\Ring\), or of \(R\)-modules in \({}_R\Mod\);

• a homeomorphism in \(\Top\), a homotopy equivalence in \(\HTop\), and a diffeomorphism in \(\Man\).

1. \(*_G\) is clearly a groupoid.

2. The category whose objects are all the vector spaces over a field \(k\), but where \(\Mor(V, W)\) is simply the set of isomorphisms \(V \rightarrow W\), is also a category, and is a groupoid. Similarly with groups, rings or any other category.

3. The fundamental groupoid of a topological space. If \(X\) is a topological space, we can define the category \(\mcC\) with \(\Ob \mcC = X\), and where for \(x, y \in X = \Ob \mcC\), \(\Mor(x, y)\) is the set of equivalence classes of continuous maps \(f : [0, 1] \rightarrow X\) with \(f(0) = x\) and \(f(1) = y\), where two such paths \(f, g : [0, 1] \rightarrow X\) are equivalent if there exists a continuous map \(H : [0, 1] \times [0, 1] \rightarrow X\) such that \(H(s, 0) = f(s), H(s, 1) = g(s)\), \(H(0, t) = x\) and \(H(1, t) = y\) for all \(s, t \in [0, 1]\). For two paths \(f : [0, 1] \rightarrow X\) and \(g : [0, 1] \rightarrow X\) such that \(f(1) = g(0)\), define \(g \circ f\) by \[(g \circ f)(t) = \begin{cases} f(2t), & \text{if $t \in [0, 1/2]$, and } \\ g(2t - 1), & \text{if $t \in [1/2, 1]$} \end{cases}\] (check that it is well-defined and continuous). Verify that this construction induces a well-defined composition in \(\mcC\), and that with the resulting composition, \(\mcC\) is indeed a category. This category is a groupoid (check), called the fundamental groupoid of \(X\): the inverse of \(f : [0, 1] \rightarrow X\) is the reverse path, \(g : [0, 1] \rightarrow X\) such that \(g(t) = f(1 - t)\) for \(0 \leq t \leq 1\).

1. The objects of a category are often sets equipped with additional structure (e.g., a multiplication law). Accordingly, we have various ‘forgetful functors’ \[\begin{split} \Forget : \Grp \longrightarrow \Set, \qquad \Forget : {}_R\Mod \longrightarrow \Ab, \\ \Forget : {}_R\Mod \longrightarrow \Set, \qquad \Forget : \Rep_k(G) \longrightarrow \Vect_k, \end{split}\] etc. For instance, \(\Forget : \Grp \longrightarrow \Set\) assigns to each group its underlying set, and assigns to each group homomorphism \(G \rightarrow H\) the same map viewed as a map of sets.

2. \(\pi_0 : \Top \longrightarrow \Set\) assigns to each topological space its set \(\pi_0(X)\) of connected components, and to each continuous map \(f : X \rightarrow Y\) of topological spaces the induced map \(\pi_0(f) : \pi_0(X) \rightarrow \pi_0(Y)\) of connected components: it is well-defined since the image of a connected component of \(X\) under the continuous map \(f\) is connected and hence contained in a connected component of \(Y\).

3. However, we don’t have a functor \(\pi_1 : \Top \longrightarrow \Grp\): \(\pi_1\) is not assigned to a topological space \(X\), but to a pointed topological space or a based topological space \((X, x)\), where \(X\) is a topological space and \(x \in X\) is a point. There is a category of pointed topological spaces, say \(\widetilde{\Top}\), where \(\Mor((X, x), (Y, y))\) is the set of continuous maps \(f : X \rightarrow Y\) such that \(f(x) = y\). Such a map \(f\) uniquely determines a group homomorphism \(\pi_1(f) : \pi_1(X, x) \rightarrow \pi_1(Y, y)\). This respects the composition and the identity morphisms, so the assignments \((X,x) \mapsto \pi_1(X,x)\) and \(f \mapsto \pi_1(f)\) define a functor \(\pi_1 : \widetilde{\Top} \rightarrow \Grp\).

   Another way to look at this is the following: if \(X\) is a nonempty path-connected topological space, changing the base-point gives an isomorphism between the corresponding fundamental groups, but this isomorphism is well-defined only up to an inner automorphism. This ambiguity makes \(\pi_1\) non-functorial. However (after choosing a base-point \(x_X\) for each nonempty path connected topological space \(X\) to fix definitions), \(\pi_1\) defines for us a functor \(\pi_1 : \Top^{\mathrm{pc}, \neq \emptyset} \longrightarrow \overline{\Grp}\), where \(\overline{\Grp}\) is the coarser category defined in Example ↗(iv), and \(\Top^{\mathrm{pc}, \neq \emptyset}\) is the category of nonempty path connected topological spaces. The resulting functor is independent of these choices up to a canonical natural isomorphism (a notion we will see soon).

4. A functor \(F : *_G \longrightarrow \Set\) is simply a set with an action of \(G\): to see this, note that \(X := F(*)\) is a set, while applying \(F\) to any \(g \in G = \Mor(*, *)\) gives \(F(g) \in \Mor_{\Set}(F(*), F(*)) = \{\text{Maps $X \rightarrow X$}\}\), and the rules \(F(g) \circ F(h) = F(g \circ h)\) and \(F(\id_*) = \id_{F(*)}\) translate to \(F(gh) = F(g) F(h)\) and that \(F(\id_*)\) is the identity map \(X \rightarrow X\). Thus, \(g \mapsto F(g)\) is a group homomorphism \(G \rightarrow \mathrm{Bij}(X, X)\), which is the same as giving an action of \(G\) on \(X\): \(g \cdot x = F(g)(x)\).

5. By the same reasoning, a functor \(F : *_G \longrightarrow \Vect_k\) is simply a representation of \(G\) on a \(k\)-vector space. More generally, a functor \(F : *_G \rightarrow \mcC\) can be thought of as an object of \(\mcC\) equipped with an action of \(G\).

6. Any group homomorphism \(G \rightarrow H\) induces a functor \(*_G \longrightarrow *_H\).

Let \(\mcC\) be a category. Every \(X \in \Ob \mcC\) determines two functors:

1. \(h_X := \Mor_{\mcC}(X, -) : \mcC \longrightarrow \Set\), i.e., \(h_X(Y) = \Mor_{\mcC}(X, Y)\), \(h_X(f) = f \circ -\).

2. \(h^X := \Mor_{\mcC}(-, X) : \mcC^{op} \longrightarrow \Set\), i.e., \(h^X(Y) = \Mor_{\mcC}(Y, X)\), \(h^X(f) = - \circ f\).

In fact, if \(G = \Aut(X)\), these functors can be upgraded to be valued in \(G\)-\(\Set\) rather than \(\Set\): \(g \in G\) acts by pre-composition with \(g^{-1}\) on \(h_X(Y)\), and by post-composition with \(g\) on \(h^X(Y)\).

1. The forgetful functors \[\Grp\longrightarrow\Set,\qquad {}_R\Mod\longrightarrow\Ab,\qquad {}_R\Mod\longrightarrow\Set,\qquad \Top\longrightarrow\Set\] are all faithful, but none of them is full. The obvious inclusion functor \(\Ab\longrightarrow\Grp\) is fully faithful.

2. If \(G \rightarrow H\) is a group homomorphism, the functor \(*_G \longrightarrow *_H\) discussed in Example ↗(vi) is faithful (resp., full; resp., fully faithful) if and only if \(G \rightarrow H\) is injective (resp., surjective; resp., bijective).

3. Consider the category \(\Vect_k^{\fd}\) of finite-dimensional \(k\)-vector spaces and \(k\)-linear transformations, and its full subcategory \(\mcC\) consisting of vector spaces of the form \(k^n\) for some \(n \in \ZZ_{\geq 0}\): this means that the members of \(\Ob \mcC\) are simply the \(k\)-vector spaces of the form \(k^n\), and that \(\Mor_{\mcC}(X, Y) = \Mor_{\Vect_k^{\fd}}(X, Y)\) for all \(X, Y \in \Ob \mcC\). Then, by definition, the inclusion functor \(\mcC \longrightarrow \Vect_k^{\fd}\) is fully faithful. Since every finite-dimensional \(k\)-vector space is isomorphic to some \(k^n\), this functor is also essentially surjective, and hence is an equivalence of categories. Note that \(\mcC\) is small, while \(\Vect_k^{\fd}\) is not.

4. \(\Vect_k^{\fd}\) is equivalent to \((\Vect_k^{\fd})^{\mathrm{op}}\), by the functor that takes \(V\) in \(\Ob \Vect_k^{\fd}\) to its dual \(V^{\vee} := \Hom_k(V, k)\), and each linear map \(T : V \rightarrow W\) to the transpose (or “pull-back under \(T\)”) map \(\,^tT : W^{\vee} \rightarrow V^{\vee}\), thought of as an element of \(\Mor_{(\Vect_k^{\fd})^{\mathrm{op}}}(V^{\vee}, W^{\vee})\).

5. Later, we will see that for any integer \(n \geq 1\), \(\Vect_k^{\fd}\) is equivalent to ‘\({}_{\M_n(k)}\Mod^{\fg}\)’, by a functor that, at the level of objects, takes \(V\) to \(h_{k^n}(V) := \Hom_k(k^n, V)\), viewed as a module over \(\End_k(k^n)^{\mathrm{op}} \cong \M_n(k)\); note that this is only a slight variant of the description in Example ↗(i). This is an example of what is known as a Morita equivalence.

6. We will see in the second half of this course that the fundamental theorem of Galois theory can be formulated as an equivalence of categories. Namely, if \(K/k\) is a finite Galois extension, then we have an equivalence of categories: \[\label{eqn: Galois theory functor} h^K : \{\text{Finite separable $k$-algebras that are $K$-split}\} \longrightarrow (\Gal(K/k)\text{-}\FinSet)^{\mathrm{op}},\] where \(\Gal(K/k)\text{-}\FinSet\) is the category of finite sets with a \(\Gal(K/k)\)-action.

   In fact, this equivalence is defined exactly as in Example ↗(ii): \(\Gal(K/k)\) is simply the automorphism group of the object \(X := K\) of the source category, and \(h^K(A) = h^X(A) := \Hom_{\Alg_k}(A, K)\) is viewed as a set with a \(\Gal(K/k)\)-action. An example of a finite separable \(k\)-algebra that is \(K\)-split is a finite separable extension of \(k\) contained in \(K\). At first sight, this can look like a bizarre complication, but this formulation turns out to be natural, with many uses. The finiteness requirement on \(K/k\) can be removed by using a continuity condition.

7. In functional analysis, the Gelfand-Naimark theorem for commutative \(C^*\)-algebras gives an equivalence between the opposite category \(\CHaus^{\mathrm{op}}\) of the category \(\CHaus\) of compact Hausdorff spaces, and the category of commutative unital \(C^*\)-algebras (where the homomorphisms are required to be unital \(*\)-homomorphisms).

8. Here are some categories from algebraic geometry that you may encounter in later courses. In algebraic geometry, given a commutative ring \(R\), the category of affine schemes over \(R\) (or over “\(\Spec R\)”) is equivalent to the opposite category \((\Alg_R)^{\mathrm{op}}\) of the category of commutative \(R\)-algebras.

   Given a commutative ring \(R\), you will study the category of affine schemes of finite type over \(R\). This category is equivalent to the opposite category \((\Alg_R^{\ft})^{\mathrm{op}}\).

   Now suppose that \(R\) is a field \(k\), say algebraically closed. If we instead consider the subcategory \(\Alg_k^{\ft, \red}\) of \(\Alg_k^{\ft}\) consisting of finite type \(k\)-algebras that are reduced (i.e., having no nilpotent elements), then its opposite category \((\Alg_k^{\ft, \red})^{\mathrm{op}}\) is equivalent to the category of affine algebraic sets over \(k\). If we only consider the category of finitely generated \(k\)-algebras that are integral domains, its opposite is equivalent to the category of affine varieties over \(k\).

   One can enlarge these categories by ‘patching their objects’ to obtain the categories of (not necessarily affine) schemes over \(R\), algebraic schemes over \(k\) and algebraic varieties over \(k\).

Recall Example ↗, parts (iv) and (v): expanding on the reasoning there, it follows that the category \(\Fun(*_G, \Set)\) can be identified with the category \(G\text{-}\Set\) of sets with a \(G\)-action (Example ↗(xv)), and the category \(\Fun(*_G, \Vect_k)\) with \(\Rep_k(G)\).

Given a category \(\mcC\), there will be two categories of interest for the next lecture: the category \(\Presh(\mcC) = \Fun(\mcC^{\mathrm{op}}, \Set)\) and the category \(\Presh(\mcC^{\mathrm{op}}) = \Fun(\mcC, \Set)\).

The following categories have arbitrary small products, which coincide with what you have already seen called the products of their objects: \(\Set, \Grp, \Top, \Ab, {}_R\Mod, \Vect_k, \Ring\). For instance, if \((X_i)_{i \in I}\) is a family of topological spaces, we take \(\prod_{i \in I} X_i\) to be the set-theoretic product of the \(X_i\), given the product topology, and \(\pi_j : \prod_{i \in I} X_i \rightarrow X_j\) to be the projection onto the \(j\)-th factor. We also need to account for the empty product, namely a terminal object, which exists in each of these cases: a singleton set \(\{*\}\) for \(\Set\), a trivial group for \(\Grp\) and \(\Ab\), a singleton topological space for \(\Top\), the zero module \(0\) for \({}_R\Mod\) (and similarly with \(\Vect_k\)), and the zero ring for \(\Ring\).

For example, for any topological space \(Y\), giving a continuous map \(f_i : Y \rightarrow X_i\) for each \(i\) is equivalent to giving a single continuous map \(f : Y \rightarrow \prod_i X_i\), such that for each \(i\), \(f\) projects along the \(i\)-th factor to \(f_i\). This is why the product topology was defined the way it was: the familiar basic open sets, etc. By the uniqueness of products (Exercise ↗), this was the only way the product topological space could have been defined.

For those of you who are familiar with algebraic varieties, the category of algebraic varieties over \(k\) has binary products, which is the ‘usual’ binary product of algebraic varieties.

The following categories have arbitrary small coproducts:

• In \(\Set\), coproduct is given by the disjoint union: \(\coprod_i X_i\) can be taken to be the disjoint union \(X\) of the \(X_i\), and \(\iota_j : X_j \rightarrow X\) to be the obvious inclusion. Of course, one also needs to remark that \(\Set\) does have an initial object, which is \(\emptyset\).

• In \(\Grp\), coproduct is given by the free product: the free product \(G * H\) of \(G\) and \(H\) consists of words \(s_1 \cdots s_n\) with each \(s_i\) belonging to \(G\) or \(H\), modulo the obvious reductions: identity elements may be removed, and two successive terms belonging to the same group may be multiplied together. Equivalently, each element has a unique reduced expression as either the empty word or an alternating sequence of nonidentity elements of \(G\) and \(H\). The maps \(G \rightarrow G * H\) and \(H \rightarrow G * H\) are obvious. \(\Grp\) does have an initial object, the trivial group.

• For \(\Ring\), a coproduct exists, and is a “ring-theoretic free product amalgamated over \(\mathbb{Z}\)”, but we will not need an explicit construction here. \(\ZZ\) is an initial object in \(\Ring\) (the zero ring cannot be an initial object, because by definition, ring homomorphisms are required to send \(1\) to \(1\)).

• In \(\Ab, {}_R\Mod\) and \(\Vect_k\), coproduct is given by direct sum (and the trivial group or the \(0\) group or module or vector space is the initial object). Thus, in all these categories, finite coproducts and finite products can be identified with each other, though not infinite ones.

• In \(\Top\), again, coproduct is given by the disjoint union, but make sure you know how to define the ‘correct’ topology on \(\bigsqcup_i X_i\): the \(X_i \subset X\) are all open and disjoint, and the topology of each \(X_i\) coincides with the one it gets from the inclusion \(X_i \subset X\). Again, \(\emptyset\) serves as an initial object.

For example, for any topological space \(Y\), giving a continuous map \(f_i : X_i \rightarrow Y\) for each \(i\) is equivalent to giving a single continuous map \(f : \coprod_i X_i \rightarrow Y\), such that for each \(i\), \(f\) restricts to \(X_i\) as \(f_i\).

1. In \(\Set\), a morphism \(f : X \rightarrow Y\) is a monomorphism (resp., epimorphism) if and only if it is an injective (resp., surjective) function.

2. The “if” part of the analogous assertion is true in \(\Grp, \Ab, \Ring, {}_R\Mod\), \(\Vect_k\) and \(\Top\), and also in the full subcategory \(\HausTop\) of \(\Top\) consisting of the Hausdorff topological spaces. This can be viewed more category-theoretically: if \(F : \mcC \longrightarrow \Set\) is a faithful functor, then \(f : X \rightarrow Y\) is a monomorphism (resp., epimorphism) whenever the map \(F(f) : F(X) \rightarrow F(Y)\) of sets is (for these categories, this is true with \(F\) the forgetful functor to \(\Set\)).

3. However, the “only if” part, while true for \(\Grp, \Top, \Ab, {}_R\Mod\) and \(\Vect_k\) (a bit of work is needed to show this for \(\Grp\) and \(\Top\)), is not true for \(\Ring\) or \(\HausTop\): it is an easy exercise to check that every monomorphism is injective in these categories (as also in \(\Grp, \Top\)), but epimorphisms may not be surjective in \(\Ring\) or \(\HausTop\): in \(\Ring\), \(\ZZ \rightarrow \QQ\) is an epimorphism, ³ while in \(\HausTop\), any morphism with a dense image is an epimorphism (easy but good exercise).

1. In \(\Set\), an equalizer \(E\) of \(f_1, f_2 : X \rightarrow Y\) is simply the (largest) subset of \(X\) where \(f_1\) and \(f_2\) agree, while their coequalizer \(Q\) is the quotient of \(Y\) by the equivalence relation generated by the relation “\(f_1(x) \sim f_2(x)\) for all \(x \in X\)”. In \(\Top\), the prescriptions for the equalizer \(E\) and the coequalizer \(Q\) are the same as for \(\Set\), except that \(E\) should be given the induced topology from \(X\) and that \(Q\) should be given the quotient topology dictated by \(Y \rightarrow Q\).

2. In \(\Grp\), the equalizer is as in \(\Set\) (but it forms a subgroup of \(X\) and one remembers it as a subgroup, not as a subset), and the coequalizer of \(f_1, f_2 : X \rightarrow Y\) is the quotient of \(Y\) by the smallest normal subgroup of \(Y\) containing \(f_1(x) f_2(x)^{-1}\) for each \(x \in X\).

3. In \(\Ab, {}_R\Mod\) and \(\Vect_k\), the equalizer of \(f_1\) and \(f_2\) is the (inclusion of the) kernel of \(f_1 - f_2\), and their coequalizer is the cokernel of \(f_1 - f_2\) (thus, for \(\Ab\), the equalizer and the coequalizer are the same as in \(\Grp\), only the description simplifies due to abelianness).

4. In \(\Presh(\mcC) := \Fun(\mcC^{\mathrm{op}}, \Set)\), the equalizer is the “object-wise set-theoretic equalizer” (make this precise and prove it).