Chapter 0_III (suite)

Preliminaries

§8. Representable functors

8.1. Representable functors

8.1.1.

We denote by Set the category of sets. Let $\mathcal{C}$ be a category; for two objects $X$, $Y$ of $\mathcal{C}$, we set $h_X(Y)=\mathrm{Hom}(Y,X)$; for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, we denote by $h_X(u)$ the map $v\mapsto v\circ u$ from $\mathrm{Hom}(Y^{\prime },X)$ into $\mathrm{Hom}(Y,X)$. It is immediate that with these definitions $h_X:\mathcal{C}\to Set$ is a contravariant functor, that is, an object of the category, denoted $\mathrm{Hom}({\mathcal{C}}^{\circ },Set)$, of covariant functors from the category ${\mathcal{C}}^{\circ }$ dual to $\mathcal{C}$ into the category Set (T, 1.7, d) and [29].

Translator's note. We render EGA's Ens as Set throughout, the standard modern English form. The 1961 original capitalizes the category as Ens; the typographical species ("category in bold") is rendered here by the upright ASCII string Set inside backticks.

8.1.2.

Now let $w:X\to X^{\prime }$ be a morphism in $\mathcal{C}$; for every $Y\in \mathcal{C}$ and every $v\in \mathrm{Hom}(Y,X)=h_X(Y)$, we have $w\circ v\in \mathrm{Hom}(Y,X^{\prime })=h_{X^{\prime }}(Y)$; we denote by $h_w(Y)$ the map $v\mapsto w\circ v$ from $h_X(Y)$ into $h_{X^{\prime }}(Y)$. It is immediate that for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, the diagram

              h_X(u)
   h_X(Y') ────────→ h_X(Y)

   h_w(Y')│           │h_w(Y)
          ↓           ↓

   h_{X'}(Y') ───────→ h_{X'}(Y)
                h_{X'}(u)

is commutative; in other words, $h_w$ is a functorial morphism $h_X\to h_{X^{\prime }}$ (T, 1.2), or again a morphism in the category $\mathrm{Hom}({\mathcal{C}}^{\circ },Set)$ (T, 1.7, d). The definitions of $h_X$ and $h_w$ therefore constitute the definition of a canonical covariant functor

  h : 𝒞 → Hom(𝒞°, Set).                                                    (8.1.2.1)

8.1.3.

Let $X$ be an object of $\mathcal{C}$, $F$ a contravariant functor from $\mathcal{C}$ into Set (an object of $\mathrm{Hom}({\mathcal{C}}^{\circ },Set)$). Let $g:h_X\to F$ be a functorial morphism: for every $Y\in \mathcal{C}$,

$g(Y)$ is a map $h_X(Y)\to F(Y)$ such that for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, the diagram

              h_X(u)
   h_X(Y') ────────→ h_X(Y)

   g(Y')  │           │ g(Y)                                              (8.1.3.1)
          ↓           ↓

   F(Y') ────────────→ F(Y)
                F(u)

is commutative. In particular, we have a map $g(X):h_X(X)=\mathrm{Hom}(X,X)\to F(X)$, whence an element

$$\alpha (g)=(g(X))(1_X)\in F(X)(\mathrm{8.1.3.2})$$

and consequently a canonical map

  α : Hom(h_X, F) → F(X).                                                  (8.1.3.3)

Conversely, consider an element $\xi \in F(X)$; for every morphism $v:Y\to X$ in $\mathcal{C}$, $F(v)$ is a map $F(X)\to F(Y)$; consider the map

$$v\mapsto (F(v))(\xi )(\mathrm{8.1.3.4})$$

from $h_X(Y)$ into $F(Y)$; if we denote this map by $(\beta (\xi ))(Y)$,

$$\beta (\xi ):h_X\to F(\mathrm{8.1.3.5})$$

is a functorial morphism, for we have, for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, $(F(v\circ u))(\xi )=(F(u)\circ F(v))(\xi )$, which verifies the commutativity of (8.1.3.1) for $g=\beta (\xi )$. We have thus defined a canonical map

  β : F(X) → Hom(h_X, F).                                                  (8.1.3.6)

Proposition (8.1.4).

The maps $\alpha$ and $\beta$ are mutually inverse bijections.

Proof. Let us compute $\alpha (\beta (\xi ))$ for $\xi \in F(X)$; for every $Y\in \mathcal{C}$, $(\beta (\xi ))(Y)$ is the map $g_1(Y):v\mapsto (F(v))(\xi )$ from $h_X(Y)$ into $F(Y)$. We therefore have

  α(β(ξ)) = (g_1(X))(1_X) = (F(1_X))(ξ) = 1_{F(X)}(ξ) = ξ.

Let us now compute $\beta (\alpha (g))$ for $g\in \mathrm{Hom}(h_X,F)$; for every $Y\in \mathcal{C}$, $(\beta (\alpha (g)))(Y)$ is the map $v\mapsto (F(v))((g(X))(1_X))$; by the commutativity of (8.1.3.1), this map is none other than $v\mapsto (g(Y))((h_X(v))(1_X))=(g(Y))(v)$ by the definition of $h_X(v)$; in other words, it is equal to $g(Y)$, which proves the proposition.

8.1.5.

Recall that a subcategory ${\mathcal{C}}^{\prime }$ of a category $\mathcal{C}$ is defined by the condition that its objects be objects of $\mathcal{C}$, and that if $X^{\prime }$, $Y^{\prime }$ are two objects of ${\mathcal{C}}^{\prime }$, the set ${\mathrm{Hom}}_{{\mathcal{C}}^{\prime }}(X^{\prime },Y^{\prime })$ of morphisms $X^{\prime }\to Y^{\prime }$ in ${\mathcal{C}}^{\prime }$ is a subset of the set ${\mathrm{Hom}}_{\mathcal{C}}(X^{\prime },Y^{\prime })$ of morphisms $X^{\prime }\to Y^{\prime }$ in $\mathcal{C}$, the canonical map of "composition of morphisms"

  Hom_{𝒞'}(X', Y') × Hom_{𝒞'}(Y', Z') → Hom_{𝒞'}(X', Z')

being the restriction of the canonical map

  Hom_𝒞(X', Y') × Hom_𝒞(Y', Z') → Hom_𝒞(X', Z').

We say that ${\mathcal{C}}^{\prime }$ is a full subcategory of $\mathcal{C}$ if ${\mathrm{Hom}}_{{\mathcal{C}}^{\prime }}(X^{\prime },Y^{\prime })={\mathrm{Hom}}_{\mathcal{C}}(X^{\prime },Y^{\prime })$ for every pair of objects of ${\mathcal{C}}^{\prime }$. The subcategory ${\mathcal{C}}^{\prime \prime }$ of $\mathcal{C}$ formed by the objects of $\mathcal{C}$ isomorphic to objects of ${\mathcal{C}}^{\prime }$ is then also a full subcategory of $\mathcal{C}$, equivalent (T, 1.2) to ${\mathcal{C}}^{\prime }$, as one verifies without difficulty.

A covariant functor $F:{\mathcal{C}}_1\to {\mathcal{C}}_2$ is said to be fully faithful if, for every pair of objects X_1, Y_1 of ${\mathcal{C}}_1$, the map $u\mapsto F(u)$ from $\mathrm{Hom}(X_1,Y_1)$ into $\mathrm{Hom}(F(X_1),F(Y_1))$ is bijective; this entails that the subcategory $F({\mathcal{C}}_1)$ of ${\mathcal{C}}_2$ is full. Moreover, if two objects X_1, $X_1^{\prime }$ have the same image X_2, there exists a unique isomorphism $u:X_1\to X_1^{\prime }$ such that $F(u)=1_{X_2}$. For each object X_2 of $F({\mathcal{C}}_1)$, let $G(X_2)$ be one of the objects X_1 of ${\mathcal{C}}_1$ such that $F(X_1)=X_2$ ($G$ being defined by means of the axiom of choice); for every morphism $v:X_2\to Y_2$ in $F({\mathcal{C}}_1)$, $G(v)$ will be the unique morphism $u:G(X_2)\to G(Y_2)$ such that $F(u)=v$; $G$ is then a functor from $F({\mathcal{C}}_1)$ into ${\mathcal{C}}_1$; $F\circ G$ is the identity functor on $F({\mathcal{C}}_1)$, and what precedes shows that there exists an isomorphism of functors $\varphi :1_{{\mathcal{C}}_1}\to G\circ F$ such that $F$, $G$, $\varphi$ and the identity $1_{F({\mathcal{C}}_1)}\to F\circ G$ define an equivalence of the category ${\mathcal{C}}_1$ with the full subcategory $F({\mathcal{C}}_1)$ of ${\mathcal{C}}_2$ (T, 1.2).

8.1.6.

Apply Proposition (8.1.4) to the case where the functor $F$ is $h_{X^{\prime }}$, $X^{\prime }$ being an arbitrary object of $\mathcal{C}$; the map $\beta :\mathrm{Hom}(X,X^{\prime })\to \mathrm{Hom}(h_X,h_{X^{\prime }})$ is here none other than the map $w\mapsto h_w$ defined in (8.1.2); this map being bijective, we see, with the terminology of (8.1.5), that:

Proposition (8.1.7).

The canonical functor $h:\mathcal{C}\to \mathrm{Hom}({\mathcal{C}}^{\circ },Set)$ is fully faithful.

8.1.8.

Let $F$ be a contravariant functor from $\mathcal{C}$ into Set; we say that $F$ is representable if there exists an object $X\in \mathcal{C}$ such that $F$ is isomorphic to $h_X$; it follows from (8.1.7) that the data of an $X\in \mathcal{C}$ and an isomorphism of functors $g:h_X\to F$ determines $X$ up to unique isomorphism. Proposition (8.1.7) also means that $h$ defines an equivalence of $\mathcal{C}$ with the full subcategory of $\mathrm{Hom}({\mathcal{C}}^{\circ },Set)$ formed by the representable contravariant functors. It moreover follows from (8.1.4) that the data of a functorial morphism $g:h_X\to F$ is equivalent to that of an element $\xi \in F(X)$; to say that $g$ is an isomorphism is equivalent, for $\xi$, to the following condition: for every object $Y$ of $\mathcal{C}$ the map $v\mapsto (F(v))(\xi )$ from $\mathrm{Hom}(Y,X)$ into $F(Y)$ is bijective. When $\xi$ satisfies this condition, we shall say that the pair $(X,\xi )$ represents the representable functor $F$. By abuse of language, we shall also say that the object $X\in \mathcal{C}$ represents $F$ if there exists $\xi \in F(X)$ such that $(X,\xi )$ represents $F$, in other words if $h_X$ is isomorphic to $F$.

Let $F$, $F^{\prime }$ be two representable contravariant functors from $\mathcal{C}$ into Set, $h_X\to F$ and $h_{X^{\prime }}\to F^{\prime }$ two isomorphisms of functors. Then it follows from (8.1.6) that there is a canonical one-to-one correspondence between $\mathrm{Hom}(X,X^{\prime })$ and the set $\mathrm{Hom}(F,F^{\prime })$ of functorial morphisms $F\to F^{\prime }$.

8.1.9. Examples. I: projective limits.

The notion of representable contravariant functor covers in particular the "dual" notion of the usual notion of "solution of a

universal problem". More generally, we shall see that the notion of projective limit is a particular case of that of representable functor. Recall that in a category $\mathcal{C}$, one defines a projective system by the data of a preordered set $I$, a family $(A_{\alpha }{)}_{\alpha \in I}$ of objects of $\mathcal{C}$, and, for every pair of indices $(\alpha ,\beta )$ such that $\alpha \le \beta$, a morphism $u_{\alpha \beta }:A_{\beta }\to A_{\alpha }$. A projective limit of this system in $\mathcal{C}$ consists of an object $B$ of $\mathcal{C}$ (denoted $\lim A_{\alpha }$), and, for each $\alpha \in I$, a morphism $u_{\alpha }:B\to A_{\alpha }$, such that: 1° $u_{\alpha }=u_{\alpha \beta }\circ u_{\beta }$ for $\alpha \le \beta$; 2° For every object $X$ of $\mathcal{C}$ and every family $(v_{\alpha }{)}_{\alpha \in I}$ of morphisms $v_{\alpha }:X\to A_{\alpha }$ such that $v_{\alpha }=u_{\alpha \beta }\circ v_{\beta }$ for $\alpha \le \beta$, there exists a unique morphism $v:X\to B$ (denoted $\lim v_{\alpha }$) such that $v_{\alpha }=u_{\alpha }\circ v$ for every $\alpha \in I$ (T, 1.8). This is interpreted as follows: the $u_{\alpha \beta }$ canonically define maps

  ū_{αβ} : Hom(X, A_β) → Hom(X, A_α)

which define a projective system of sets $(\mathrm{Hom}(X,A_{\alpha }),{\bar{u}}_{\alpha \beta })$, and $(v_{\alpha })$ is by definition an element of the set $\lim \mathrm{Hom}(X,A_{\alpha })$; it is clear that $X\mapsto \lim \mathrm{Hom}(X,A_{\alpha })$ is a contravariant functor from $\mathcal{C}$ into Set, and the existence of the projective limit $B$ is equivalent to saying that $(v_{\alpha })\mapsto \lim v_{\alpha }$ is an isomorphism of functors in $X$

  lim Hom(X, A_α) ⥲ Hom(X, B)                                              (8.1.9.1)

in other words that the functor $X\mapsto \lim \mathrm{Hom}(X,A_{\alpha })$ is representable.

8.1.10. Examples. II: final object.

Let $\mathcal{C}$ be a category, $a$ a set reduced to a single element. Consider the contravariant functor $F:\mathcal{C}\to Set$ which assigns to every object $X$ of $\mathcal{C}$ the set $a$, and to every morphism $X\to X^{\prime }$ in $\mathcal{C}$ the unique map $a\to a$. To say that this functor is representable means that there exists an object $e\in \mathcal{C}$ such that for every $Y\in \mathcal{C}$, $\mathrm{Hom}(Y,e)=h_e(Y)$ is reduced to one element; we say that $e$ is a final object of $\mathcal{C}$, and it is clear that two final objects of $\mathcal{C}$ are isomorphic (which allows us to define, in general by means of the axiom of choice, a final object of $\mathcal{C}$, then denoted $e_{\mathcal{C}}$). For example, in the category Set, the final objects are the sets reduced to one element; in the category of augmented algebras over a field $K$ (where the morphisms are the algebra homomorphisms compatible with the augmentations), $K$ is a final object; in the category of $S$-preschemes (I, 2.5.1), $S$ is a final object.

8.1.11.

For two objects $X$, $Y$ of a category $\mathcal{C}$, set $h_X^{\prime }(Y)=\mathrm{Hom}(X,Y)$, and for every morphism $u:Y\to Y^{\prime }$, let $h_X^{\prime }(u)$ be the map $v\mapsto u\circ v$ from $\mathrm{Hom}(X,Y)$ into $\mathrm{Hom}(X,Y^{\prime })$; $h_X^{\prime }$ is then a covariant functor $\mathcal{C}\to Set$, from which one deduces, as in (8.1.2), the definition of a canonical covariant functor $h^{\prime }:{\mathcal{C}}^{\circ }\to \mathrm{Hom}(\mathcal{C},Set)$; a covariant functor $F$ from $\mathcal{C}$ into Set, that is, an object of $\mathrm{Hom}(\mathcal{C},Set)$, is then said to be representable if there exists an object $X\in \mathcal{C}$ (necessarily unique up to unique isomorphism) such that $F$ is isomorphic to $h_X^{\prime }$; we leave to the reader the task of developing the "dual" considerations of the preceding ones for this notion, which this time covers that of inductive limit, and in particular the usual notion of "solution of a universal problem".

8.2. Algebraic structures in categories

8.2.1.

Given two contravariant functors $F$, $F^{\prime }$ from $\mathcal{C}$ into Set, recall that for every object $Y\in \mathcal{C}$, we set $(F\times F^{\prime })(Y)=F(Y)\times F^{\prime }(Y)$, and for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, we set $(F\times F^{\prime })(u)=F(u)\times F^{\prime }(u)$, which is the map $(t,t^{\prime })\mapsto ((F(u))(t),(F^{\prime }(u))(t^{\prime }))$ from $F(Y^{\prime })\times F^{\prime }(Y^{\prime })$ into $F(Y)\times F^{\prime }(Y)$; $F\times F^{\prime }:\mathcal{C}\to Set$ is therefore a contravariant functor (which is moreover none other than the product of the objects $F$, $F^{\prime }$ in the category $\mathrm{Hom}({\mathcal{C}}^{\circ },Set)$). Given an object $X\in \mathcal{C}$, we shall call an internal composition law on $X$ a functorial morphism

  γ_X : h_X × h_X → h_X.                                                   (8.2.1.1)

In other words (T, 1.2), for every object $Y\in \mathcal{C}$, ${\gamma }_X(Y)$ is a map $h_X(Y)\times h_X(Y)\to h_X(Y)$ (so by definition an internal composition law on the set $h_X(Y)$) subject to the condition that, for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, the diagram

                    h_X(u) × h_X(u)
   h_X(Y') × h_X(Y') ──────────────→ h_X(Y) × h_X(Y)

   γ_X(Y')         │                          │ γ_X(Y)
                   ↓                          ↓

       h_X(Y') ───────────────────────→ h_X(Y)
                          h_X(u)

is commutative; this means that for the composition laws ${\gamma }_X(Y)$ and ${\gamma }_X(Y^{\prime })$, $h_X(u)$ is a homomorphism from $h_X(Y^{\prime })$ into $h_X(Y)$.

In the same way, given two objects $Z$, $X$ of $\mathcal{C}$, one calls an external composition law on $X$, having $Z$ as domain of operators, a functorial morphism

  ω_{X,Z} : h_Z × h_X → h_X.                                               (8.2.1.2)

One sees as above that for every $Y\in \mathcal{C}$, ${\omega }_{X,Z}(Y)$ is an external composition law on $h_X(Y)$, having $h_Z(Y)$ as domain of operators, and such that for every morphism $u:Y\to Y^{\prime }$, $h_Z(u)$ and $h_X(u)$ form a di-homomorphism from $(h_Z(Y^{\prime }),h_X(Y^{\prime }))$ into $(h_Z(Y),h_X(Y))$.

8.2.2.

Let $X^{\prime }$ be a second object of $\mathcal{C}$, and suppose given on $X^{\prime }$ an internal composition law ${\gamma }_{X^{\prime }}$; we shall say that a morphism $w:X\to X^{\prime }$ in $\mathcal{C}$ is a homomorphism for these composition laws, if for every $Y\in \mathcal{C}$, $h_w(Y):h_X(Y)\to h_{X^{\prime }}(Y)$ is a homomorphism for the composition laws ${\gamma }_X(Y)$ and ${\gamma }_{X^{\prime }}(Y)$. If X'' is a third

object of $\mathcal{C}$ equipped with an internal composition law ${\gamma }_{X^{\prime \prime }}$ and $w^{\prime }:X^{\prime }\to X^{\prime \prime }$ a morphism in $\mathcal{C}$ which is a homomorphism for ${\gamma }_{X^{\prime }}$ and ${\gamma }_{X^{\prime \prime }}$, it is clear that the morphism $w^{\prime }\circ w:X\to X^{\prime \prime }$ is a homomorphism for the composition laws ${\gamma }_X$ and ${\gamma }_{X^{\prime \prime }}$. An isomorphism $w:X\stackrel{\sim }{\to }{X}^{\prime }$ in $\mathcal{C}$ is called an isomorphism for the composition laws ${\gamma }_X$ and ${\gamma }_{X^{\prime }}$ if $w$ is a homomorphism for these composition laws, and if its inverse morphism $w^{-1}$ is a homomorphism for the composition laws ${\gamma }_{X^{\prime }}$ and ${\gamma }_X$.

One defines in the same way the di-homomorphisms for the pairs of objects of $\mathcal{C}$ equipped with external composition laws.

8.2.3.

When an internal composition law ${\gamma }_X$ on an object $X\in \mathcal{C}$ is such that ${\gamma }_X(Y)$ is a group law on $h_X(Y)$ for every $Y\in \mathcal{C}$, we say that $X$, equipped with this law, is a $\mathcal{C}$-group or a $\mathcal{C}$-object in groups. One defines in the same way $\mathcal{C}$-rings, $\mathcal{C}$-modules, etc.

8.2.4.

Suppose that the product $X\times X$ of an object $X\in \mathcal{C}$ by itself exists in $\mathcal{C}$; by definition, we then have $h_{X\times X}=h_X\times h_X$ up to canonical isomorphism, since this is a particular case of projective limit (8.1.9); an internal composition law on $X$ may therefore be considered as a functorial morphism ${\gamma }_X:h_{X\times X}\to h_X$, and so canonically determines (8.1.6) an element $c_X\in \mathrm{Hom}(X\times X,X)$ such that $h_{c_X}={\gamma }_X$; in this case, the data of an internal composition law on $X$ is therefore equivalent to that of a morphism $X\times X\to X$; when $\mathcal{C}$ is the category Set, one recovers the classical notion of internal composition law on a set. One has an analogous result for an external composition law when the product $Z\times X$ exists in $\mathcal{C}$.

8.2.5.

With the preceding notations, suppose in addition that $X\times X\times X$ exists in $\mathcal{C}$; the characterization of the product as an object representing a functor (8.1.9) entails the existence of canonical isomorphisms

  (X × X) × X ⥲ X × X × X ⥲ X × (X × X);

if one canonically identifies $X\times X\times X$ with $(X\times X)\times X$, the map ${\gamma }_X(Y)\times 1_{h_X(Y)}$ identifies with $h_{c_X\times 1_X}(Y)$ for every $Y\in \mathcal{C}$. It is therefore equivalent to say that for every $Y\in \mathcal{C}$, the internal law ${\gamma }_X(Y)$ is associative, or that the diagram of maps

                         γ_X(Y) × 1
   h_X(Y) × h_X(Y) × h_X(Y) ────────→ h_X(Y) × h_X(Y)

   1 × γ_X(Y) │                              │ γ_X(Y)
              ↓                              ↓

   h_X(Y) × h_X(Y) ──────────────────────→ h_X(Y)
                          γ_X(Y)

is commutative, or that the diagram of morphisms

                         c_X × 1_X
       X × X × X ────────────────→ X × X

   1_X × c_X │                       │ c_X
             ↓                       ↓

         X × X ──────────────────→ X
                          c_X

is commutative.

8.2.6.

Under the hypotheses of (8.2.5), if one wants to express that for every $Y\in \mathcal{C}$, the internal law ${\gamma }_X(Y)$ is a group law, it is necessary, on the one hand, to express that it is associative, and on the other that there exists a map ${\alpha }_X(Y):h_X(Y)\to h_X(Y)$ having the properties of the inverse in a group; since for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, we have seen that $h_X(u)$ must be a group homomorphism $h_X(Y^{\prime })\to h_X(Y)$, one sees first that ${\alpha }_X:h_X\to h_X$ must be a functorial morphism. One can on the other hand express the characteristic properties of the inverse $s\mapsto s^{-1}$ in a group $G$ without making the neutral element intervene: it suffices to write that the two composed maps

  (s, t) ↦ (s, s^{−1}, t) ↦ (s, s^{−1} t) ↦ s(s^{−1} t)
  (s, t) ↦ (s, s^{−1}, t) ↦ (s, t s^{−1}) ↦ (t s^{−1}) s

are equal to the second projection $(s,t)\mapsto t$ from $G\times G$ into $G$. By virtue of (8.1.3), we have ${\alpha }_X=h_{a_X}$, where $a_X\in \mathrm{Hom}(X,X)$; the first preceding condition then expresses that the composed morphism

                  (1_X, a_X) × 1_X         1_X × c_X            c_X
   X × X ──────────────────────→ X × X × X ────────→ X × X ────────→ X

is the second projection $X\times X\to X$ in $\mathcal{C}$, and the second condition is translated similarly.

8.2.7.

Suppose now that there exists in $\mathcal{C}$ a final object $e$ (8.1.10). Suppose still that ${\gamma }_X(Y)$ is a group law on $h_X(Y)$ for every $Y\in \mathcal{C}$, and denote by ${\eta }_X(Y)$ the neutral element of ${\gamma }_X(Y)$. Since, for every morphism $u:Y\to Y^{\prime }$ in $\mathcal{C}$, $h_X(u)$ is a group homomorphism, we have ${\eta }_X(Y)=(h_X(u))({\eta }_X(Y^{\prime }))$; taking in particular $Y^{\prime }=e$, in which case $u$ is the unique element $ϵ$ of $\mathrm{Hom}(Y,e)$, we see that the element ${\eta }_X(e)$ completely determines ${\eta }_X(Y)$ for every $Y\in \mathcal{C}$. Set $e_X={\eta }_X(X)$, the neutral element of the group $h_X(X)=\mathrm{Hom}(X,X)$; the commutativity of the diagram

              h_ε(X)
   h_X(e) ───────────→ h_X(Y)

   h_{e_X}(e) │           │ h_{e_X}(Y)
              ↓           ↓

   h_X(e) ───────────→ h_X(Y)
              h_ε(X)

(cf. 8.1.2) shows that, in the set $h_X(Y)$, the map $h_{e_X}(Y)$ is none other than

$s\mapsto {\eta }_X(Y)$, transforming every element into the neutral element. One then verifies that the fact that ${\eta }_X(Y)$ is the neutral element of ${\gamma }_X(Y)$ for every $Y\in \mathcal{C}$ is equivalent to saying that the composed morphism

              (1_X, 1_X)         1_X × e_X            c_X
   X ──────────────────→ X × X ──────────→ X × X ──────→ X,

and the analogous one where one permutes 1_X and $e_X$, are both equal to 1_X.

8.2.8.

One could of course multiply without difficulty the examples of algebraic structures in categories. The example of groups has been treated with sufficient detail, but in what follows we shall generally leave to the reader the task of developing analogous considerations in the examples of algebraic structures that we shall encounter.

Translator's footnote (from p. 5). To facilitate reference lookup, we shall henceforth refer to the paragraphs of Chapter 0 published with Chapter I by the sign 0_I.