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SGA 4-I — Expose I: Presheaves

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Expose I — Presheaves

By A. Grothendieck and J.-L. Verdier, with an appendix by N. Bourbaki.

In sections 0 to 5 of this expose, we present the elementary, and most often well-known, properties of categories of presheaves (*). These properties are used constantly in the sequel of the Seminar, and knowing them is essential for understanding the following exposes. The proofs are immediate; they are most often omitted.

In sections 6 to 9 we touch on a few themes used several times later. The reader in a hurry may omit them on first reading. Section 10 fixes the terminology used. Appendix 11 is due to N. Bourbaki.

(*) The reader may also consult SGA 3 I, §§ 1 to 3.

0. Universes

A universe is a nonempty set $U$ having the following properties:

• (U1) If $x\in U$ and if $y\in x$, then $y\in U$.
• (U2) If $x,y\in U$, then $\{x,y\}\in U$.
• (U3) If $x\in U$, then $\mathcal{P}(x)\in U$.
• (U4) If $(x_i{)}_{i\in I}$ is a family of elements of $U$ and if $I\in U$, then ${\bigcup }_{i\in I}{x}_i\in U$.

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From the preceding axioms one easily deduces the following properties:

• If $x\in U$, the set $\{x\}$ belongs to $U$.
• If $x$ is a subset of $y\in U$, then $x\in U$.
• If $x,y\in U$, the ordered pair $(x,y)=\{\{x,y\},\{x\}\}$ (Kuratowski’s definition) is an element of $U$.
• If $x,y\in U$, the union $x\cup y$ and the product $x\times y$ are elements of $U$.
• If $(x_i{)}_{i\in I}$ is a family of elements of $U$ and if $I\in U$, the product ${\prod }_{i\in I}{x}_i$ is an element of $U$.
• If $x\in U$, then $\mathrm{card}(\mathcal{P}(x))<\mathrm{card}(U)$, strictly. In particular the relation $U\in U$ does not hold.

Thus one can perform all the usual operations of set theory starting from the elements of a universe without the final result thereby ceasing to be an element of the universe.

The first use of the notion of universe is to provide a definition of the usual categories: the category of sets belonging to the universe $U$, denoted $U\text{-}\mathbf{Ens}$; the category of topological spaces belonging to $U$; the category of commutative groups belonging to $U$, denoted $U\text{-}\mathbf{Ab}$; the category of categories belonging to $U$; and so on.

However, the only known universe is the set of symbols of the form $\varnothing$, $\{\varnothing \}$, $\{\{\varnothing \},\varnothing \}$, and so on. All elements of this universe are finite sets, and this universe is countable. In particular, no universe is known which contains an element of infinite cardinality. One is therefore led to add to the axioms of set theory the axiom:

$(\mathrm{UA})$ For every set $x$, there exists a universe $U$ such that $x\in U$.

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Since the intersection of a family of universes is a universe, it follows immediately that every set is an element of a smallest universe. One can show that axiom (UA) is independent of the axioms of set theory.

We shall also add the axiom:

$(\mathrm{UB})$ Let $R(x)$ be a relation and let $U$ be a universe. If there exists an element $y\in U$ such that $R(y)$, then there exists $x\in U$ such that $R(x)$.

The non-contradiction of the axioms (UA) and (UB), relative to the other axioms of set theory, has not been proved and, it seems, is not provable.

Let $U$ be a universe and let $c(U)$ be the least upper bound of the cardinals of the elements of $U$, so that $c(U)=\mathrm{card}(U)$. The cardinal $c(U)$ has the following properties:

• (FI) If $a<c(U)$, then $2^a<c(U)$.
• (FII) If $(a_i{)}_{i\in I}$ is a family of cardinals strictly smaller than $c(U)$, and if $\mathrm{card}(I)$ is strictly smaller than $c(U)$, then ${\sum }_{i\in I}{a}_i<c(U)$.

Cardinals which possess properties (FI) and (FII) are called strongly inaccessible cardinals. The cardinal $0$ and the countably infinite cardinal are strongly inaccessible.

Axiom (UA) implies:

$({\mathrm{UA}}^{\prime })$ Every cardinal is strictly bounded above by a strongly inaccessible cardinal.

Conversely, one can show that the non-contradiction of $({\mathrm{UA}}^{\prime })$ implies the non-contradiction of (UA), and that the non-contradiction of these axioms entails that of axiom (UB).

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Call a set $E$ artinian if there is no infinite family $(x_n{)}_{n\in \mathbf{N}}$ such that $x_{n+1}\in x_n\in E$. One can then show that there is a one-to-one correspondence between strongly inaccessible cardinals and artinian universes: to every strongly inaccessible cardinal $c$ one associates the unique artinian universe $U$ such that $\mathrm{card}(U)=c$.

Let $c<c^{\prime }$ be two strongly inaccessible cardinals. Then $U_c\in U_{c^{\prime }}$. In particular, the artinian universes of cardinals smaller than a given cardinal form a well-ordered set for the membership relation.

Axiom $({\mathrm{UA}}^{\prime })$ is equivalent to the axiom:

$({\mathrm{UA}}^{\prime \prime })$ Every artinian set is an element of an artinian universe.

Note that all usual sets, such as $\mathbf{Z}$, $\mathbf{Q}$, and so on, are artinian sets.

1. $U$-Categories. Presheaves of Sets

1.0. In the sequel of the Seminar, and unless expressly stated otherwise, the universes considered will contain an element of infinite cardinality. Let $U$ be a universe. A set is said to be $U$-small, or simply small when this causes no confusion, if it is isomorphic to an element of $U$. We also use the terminology: small group, small ring, small category, and so on. We shall often suppose, without explicit mention, that the schemes, topological spaces, and index sets with which we work are elements of $U$,

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or at least have cardinality in $U$. However, many categories with which we shall work will not be elements of $U$.

Definition 1.1. Let $U$ be a universe and let $C$ be a category. We say that $C$ is a $U$-category if, for every pair $(x,y)$ of objects of $C$, the set ${\mathrm{Hom}}_C(x,y)$ is $U$-small.

1.1.1. Let $C$ and $D$ be two categories, and let $\mathrm{Fonct}(C,D)$ be the category of functors from $C$ to $D$. The following assertions are checked immediately:

a) If $C$ and $D$ are elements of a universe $U$, respectively $U$-small, then the category $\mathrm{Fonct}(C,D)$ is an element of $U$, respectively is $U$-small.

b) If $C$ is $U$-small and if $D$ is a $U$-category, then $\mathrm{Fonct}(C,D)$ is a $U$-category.

Remark 1.1.2. Let $D$ be a category having the following properties:

• (C1) The set $\mathrm{ob}(D)$ is contained in the universe $U$.
• (C2) For every pair $(x,y)$ of objects of $D$, the set ${\mathrm{Hom}}_D(x,y)$ is an element of $U$.

The usual categories constructed from a universe $U$ have these two properties: $U\text{-}\mathbf{Ens}$, $U\text{-}\mathbf{Ab}$, and so on. Let $C$ be a category belonging to $U$. Then the category $\mathrm{Fonct}(C,D)$ does not in general have properties (C1) and (C2). For example the category $\mathrm{Fonct}(C,U\text{-}\mathbf{Ens})$ has neither property (C1) nor property (C2). This justifies the adopted definition of $U$-category, in preference to the more restrictive notion given by conditions (C1) and (C2).

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Definition 1.2. Let $C$ be a category. The category of presheaves of sets on $C$ relative to the universe $U$, or simply the category of presheaves on $C$ when no confusion can result, is the category of contravariant functors on $C$ with values in the category of $U$-sets.

We denote by $\hat{C}$ the category of presheaves of sets on $C$ relative to the universe $U$. The objects of $\hat{C}$ are called $U$-presheaves, or more simply presheaves, on $C$. When $C$ is $U$-small, the category $\hat{C}$ is a $U$-category. When $C$ is a $U$-category, $\hat{C}$ is not necessarily a $U$-category.

Construction-Definition 1.3. Let $x$ be an object of a $U$-category $C$. The $U$-functor represented by $x$ is the functor $h_U(x):C^{\circ }\to U\text{-}\mathbf{Ens}$ whose construction follows (*). Let $y$ be an object of $C$.

a) If ${\mathrm{Hom}}_C(y,x)$ is an element of $U$, then $h_U(x)(y)={\mathrm{Hom}}_C(y,x)$.

b) Suppose that ${\mathrm{Hom}}_C(y,x)$ is not an element of $U$. Let $R(Z,x,y)$ be the relation: “$Z$ is the target of an isomorphism ${\mathrm{Hom}}_C(y,x)\to Z$.” We then put

$$h_U(x)(y)=\tau ZR(Z,x,y).$$

By axiom (UB), $h_U(x)(y)$ is an element of $U$. Let $R^{\prime }(u,x,y)$ be the relation: “$u$ is a bijection ${\mathrm{Hom}}_C(y,x)\to h_U(x)(y)$.” We then put

$$\alpha (y,x)=\tau u{R}^{\prime }(u,x,y).$$

(*) $C^{\circ }$ will always denote the category opposite to $C$.

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Observe that in cases (a) and (b) one has a canonical isomorphism

$$\alpha (y,x):{\mathrm{Hom}}_C(y,x)\to h_U(x)(y).$$

In case (a), $\alpha (y,x)$ is the identity.

Let $u:y\to y^{\prime }$ be an arrow of $C$. By composition of morphisms, the morphism $u$ defines a map

$${\mathrm{Hom}}_C(y^{\prime },x)\to {\mathrm{Hom}}_C(y,x).$$

We put

$$h_U(x)(u)=\alpha (y,x)\circ {\mathrm{Hom}}_C(u,x)\circ \alpha (y^{\prime },x{)}^{-1}.$$

It is checked immediately that $h_U(x)$, so defined, is a functor $C^{\circ }\to U\text{-}\mathbf{Ens}$.

1.3.1. Similarly, using the isomorphisms $\alpha$, for every morphism $\gamma :x\to x^{\prime }$ one defines a morphism of functors

$$h_U(\gamma ):h_U(x)\to h_U(x^{\prime }),$$

and it is checked immediately that this defines a functor

$$h_U:C\to \hat{C}.$$

1.3.2. Now let $V$ be a universe such that $U\in V$. Then there is a canonical fully faithful functor

$$U\text{-}\mathbf{Ens}\to V\text{-}\mathbf{Ens},$$

hence a fully faithful functor

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$${\hat{C}}_U\to {\hat{C}}_V,$$

and the diagram

C -> C^_U
|    |
v    v
C -> C^_V

is commutative up to canonical isomorphism. When $C$ is an element of $U$, this canonical isomorphism is the identity.

1.3.3. In practice, the universe $U$ is fixed once and for all and is not mentioned. We then use the notation $\hat{C}$ for the category of $U$-presheaves of sets, and

$$h:C\to \hat{C}.$$

For every object $x$ of $C$, the presheaf $h(x)$ is called the presheaf represented by $x$, and we shall always identify the value at $y\in \mathrm{ob}(C)$ of the presheaf $h(x)$ with ${\mathrm{Hom}}_C(y,x)$.

Proposition 1.4. Let $C$ be a $U$-category, let $F$ be a presheaf on $C$, and let $X$ be an object of $C$. There is an isomorphism, functorial in $X$ and in $F$:

$${\mathrm{Hom}}_{\hat{C}}(h(X),F)\simeq F(X).$$

When $F$ is of the form $h(Y)$, this isomorphism is precisely the map

$${\mathrm{Hom}}_C(X,Y)\to {\mathrm{Hom}}_{\hat{C}}(h(X),h(Y)).$$

In particular the functor $h$ is fully faithful.

1.4.1. This proposition justifies the usual abuses of language identifying an object of $C$ and the corresponding contravariant functor.

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A presheaf isomorphic to an object in the image of $h$, or, using the abuse of language just mentioned, isomorphic to an object of $C$, is called a representable presheaf.

1.4.2. Let $V$ be a universe containing a universe $U$. The category of $U$-presheaves is a full subcategory of the category of $V$-presheaves. Consequently a $U$-presheaf is representable if and only if its image in the category of $V$-presheaves is a representable $V$-presheaf.

2. Projective and Inductive Limits

Let $C$ be a $U$-category and let $I$ be a small category. Denote by $\mathrm{Fonct}(I,C)$ the category of functors from $I$ to $C$. The category $\mathrm{Fonct}(I,C)$ is a $U$-category.

To every object $X$ of $C$, associate the subcategory $X$ of $C$ having as its only object the object $X$ and as its only arrow the identity of $X$. Let $i_X:X\to C$ denote the inclusion functor. There is one and only one functor $e_X:I\to X$, and we shall denote by

$$k_X:I\to C$$

the functor $i_X\circ e_X$. We shall say that $k_X$ is the constant functor with value $X$. The correspondence $X\mapsto k_X$ is visibly functorial in $X$, which allows us to define, for every functor $G:I\to C$, the presheaf

$$X\mapsto {\mathrm{Hom}}_{\mathrm{Fonct}(I,C)}(k_X,G).$$

Definition 2.1. The projective limit of $G$, denoted ${\underleftarrow{\lim }}_{I}G$, is the presheaf

$$X\mapsto {\mathrm{Hom}}_{\mathrm{Fonct}(I,C)}(k_X,G).$$

If this presheaf is representable, we again denote by ${\underleftarrow{\lim }}_{I}G$

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an object of $C$ which represents it. The object ${\underleftarrow{\lim }}_{I}G$ is therefore defined only up to isomorphism. When no confusion can result, we use the abbreviated notation $\underleftarrow{\lim }G$.

For variable $G$, ${\underleftarrow{\lim }}_{I}G$ is a functor from the category $\mathrm{Fonct}(I,C)$ with values in $\hat{C}$.

2.1.1. Similarly, by symmetry, reversing the direction of the arrows in $C$, one defines the inductive limit of a functor: it is a covariant functor on $C$, with values in the category of $U$-sets. We shall use the notations ${\underrightarrow{\lim }}_{I}G$ or $\underrightarrow{\lim }G$.

We note that products, fiber products, and kernels are projective limits. Similarly, sums, amalgamated sums, and cokernels are inductive limits.

Definition 2.2. Let $I$ and $C$ be two categories, and let $G:I\to C$ be a functor. We say that the projective limit of $G$ is representable if there exists a universe $U$ such that:

1. The category $I$ is $U$-small.
2. The category $C$ is a $U$-category.
3. The presheaf ${\underleftarrow{\lim }}_{I}G$, with values in the category of $U$-sets, is representable.

It follows from no. 1 that the object ${\underleftarrow{\lim }}_{I}G$ representing the presheaf ${\underleftarrow{\lim }}_{I}G$ does not depend, up to isomorphism, on the universe $U$. Note also that there exists a smallest universe $U^{\prime }$ having properties 1 and 2, and that the presheaf ${\underleftarrow{\lim }}_{I}G$ necessarily has values in the category of $U^{\prime }$-sets.

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2.2.1. Let $I$ and $C$ be two categories. We say that the $I$-projective limits in $C$ are representable if, for every functor $G:I\to C$, the projective limit of $G$ is representable.

Finally, let $C$ be a category and let $U$ be a universe. We say that the $U$-projective limits in $C$ are representable if, for every $U$-small category $I$ and every functor $G:I\to C$, the projective limit of $G$ is representable.

Proposition 2.3. Let $C$ be a category and let $U$ be a nonempty universe. The following assertions are equivalent:

1. The $U$-projective limits in $C$ are representable.
2. Products indexed by a small set are representable, and kernels of pairs of arrows are representable.
3. Products indexed by a small set are representable, and fiber products are representable.

Proof. It suffices to observe that there is an isomorphism functorial in $G$,

$$\underset{I}{\underleftarrow{\lim }}G\simeq \mathrm{Ker}\left(\prod _{i\in \mathrm{ob}(I)}G(i)\rightrightarrows \prod _{u\in \mathrm{Fl}(I)}G(\mathrm{target}(u))\right),$$

the pair of arrows being defined by the morphisms

$$\prod _{i\in \mathrm{ob}(I)}G(i)\stackrel{{\mathrm{pr}}_{\mathrm{target}(u)}}{\to }G(\mathrm{target}(u)),$$

$$\prod _{i\in \mathrm{ob}(I)}G(i)\stackrel{{\mathrm{pr}}_{\mathrm{source}(u)}}{\to }G(\mathrm{source}(u))\stackrel{G(u)}{\to }G(\mathrm{target}(u)).$$

Moreover it is clear that $\mathrm{Ker}(X\rightrightarrows Y)=X{\times }_{Y\times Y}Y$, the two morphisms from $X$ to $Y\times Y$ being ${\mathrm{id}}_X\times u$ and ${\mathrm{id}}_X\times v$.

2.3.1. There are of course analogous definitions and assertions for inductive limits; we shall not spell them out. Similarly for finite projective and inductive limits, that is, those relative to a finite category $I$.

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Corollary 2.3.2. Let $U\text{-}\mathbf{Ens}$ denote the category of $U$-sets and $U\text{-}\mathbf{Ab}$ the category of abelian groups in $U\text{-}\mathbf{Ens}$. The $U$-projective and inductive limits in $U\text{-}\mathbf{Ens}$ and in $U\text{-}\mathbf{Ab}$ are representable.

Proposition 2.3.3. Let $I$ be a small category, let $F:I\to U\text{-}\mathbf{Ens}$ be a functor, and let $G\subset \mathrm{ob}(I)$ be a small set of objects of $I$ such that, for every object $X$ of $I$, there exists an object $Y\in G$ and a morphism $Y\to X$ (resp. $X\to Y$). Then ${\underleftarrow{\lim }}_{I}F$ (resp. ${\underrightarrow{\lim }}_{I}F$) is representable and one has

$$\mathrm{card}(\underset{I}{\underleftarrow{\lim }}F)\le \mathrm{card}\left(\prod _{Y\in G}F(Y)\right),$$

respectively

$$\mathrm{card}(\underset{I}{\underrightarrow{\lim }}F)\le \sum _{Y\in G}\mathrm{card}(F(Y)).$$

Proof. It is checked immediately that ${\underleftarrow{\lim }}_{I}F$ (resp. ${\underrightarrow{\lim }}_{I}F$) is isomorphic to a subobject (resp. a quotient) of ${\prod }_{Y\in G}F(Y)$ (resp. of ${\coprod }_{Y\in G}F(Y)$), whence the proposition.

Definition 2.4.1. Let $C$ be a category in which finite projective (resp. inductive) limits are representable, and let $u:C\to C^{\prime }$ be a functor. We say that $u$ is left exact (resp. right exact) if it commutes with finite projective (resp. inductive) limits. A functor which is both left and right exact is called an exact functor.

2.4.2. It follows from 2.3 that, for a functor to be left exact, it is necessary and sufficient that it transform the final object, that is, the empty product, into the final object, the product of two objects into the product of their two image objects, and the kernel of pairs of two arrows into the kernel of the image pairs. Equivalently, it is necessary and sufficient that it transform the final object into the final object and fiber products into fiber products, assuming that in $C$ finite limits are representable.

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2.5.0. Let $I$, $J$, and $C$ be three categories, and let $G:I\times J\to C$ be a functor, that is, a functor from $I$ with values in the category of functors from $J$ to $C$. Suppose that the projective limits of the functors

$$G_i:J\to C,j\mapsto G(i,j)(i\in \mathrm{ob}I)$$

are representable, and that the functor

$$i\mapsto \underset{J}{\underleftarrow{\lim }}{G}_i$$

admits a representable projective limit. It is clear that the functor $G$ then admits a representable projective limit and that one has a canonical isomorphism

$$\underset{I\times J}{\underleftarrow{\lim }}G\simeq \underset{I}{\underleftarrow{\lim }}\underset{J}{\underleftarrow{\lim }}{G}_i.$$

One deduces a canonical isomorphism

$$\underset{I}{\underleftarrow{\lim }}\underset{J}{\underleftarrow{\lim }}{G}_i\simeq \underset{J}{\underleftarrow{\lim }}\underset{I}{\underleftarrow{\lim }}{G}^j.$$

In the sequel, more briefly, we shall say that projective limits commute with projective limits. Similarly, inductive limits commute with inductive limits. But it is not true in general that inductive limits commute with projective limits.

Definition 2.5. Let $C$ be a category with representable fiber products, let $I$ be a category, let $G:I\to C$ be a functor, let $g:G\to X$ be a morphism from $G$ to an object of $C$, that is, a morphism from $G$ to the constant functor associated with $X$, and let $m:Y\to X$ be a morphism of $C$.

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Let $G{\times }_{X}Y:I\to C$ be the functor obtained by base change. We say that the inductive limit of $G$ is universal if, for every object $X$, every morphism $g:G\to X$, and every morphism $m:Y\to X$:

1. the inductive limit of the functor $G{\times }_{X}Y$ is representable;

2. the canonical morphism

   $$\underset{I}{\underrightarrow{\lim }}(G{\times }_{X}Y)\to (\underset{I}{\underrightarrow{\lim }}G){\times }_{X}Y$$

   is an isomorphism.

Proposition 2.6. Let $U$ be a universe. Inductive limits in $U\text{-}\mathbf{Ens}$ which are representable, in particular $U$-inductive limits (2.2.1) in $U\text{-}\mathbf{Ens}$, are universal.

We shall also use another commutation result between projective and inductive limits, which we now present.

Definition 2.7. A category $I$ is pseudo-filtering when it has the following properties:

PS 1) Every finite diagram of two objects can be completed by an object receiving them both; in other words, for two objects $i,j$, there exists an object $k$ and morphisms $i\to k$, $j\to k$.

PS 2) For every pair of parallel arrows

$$u,v:i\rightrightarrows j,$$

there exists an arrow $w:j\to k$ such that

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$$wu=wv.$$

A category $I$ is called filtering if it is pseudo-filtering, nonempty, and connected, that is, if any two objects of $I$ can be joined by a sequence of arrows, with no condition on the direction of the arrows. In the presence of PS 2, this also means that $I$ is nonempty and that, for two objects $a,b$ of $I$, there always exists an object $c$ of $I$ and arrows $a\to c$, $b\to c$. One also says that a category $I$ is cofiltering if $I^{\circ }$ is filtering.

Example 2.7.1. If, in $I$, amalgamated sums (resp. sums of two objects) and cokernels of double arrows are representable, then $I$ is pseudo-filtering (resp. filtering).

Proposition 2.8. Let $U$ be a universe. Filtering $U$-inductive limits in $U\text{-}\mathbf{Ens}$ commute with finite projective limits.

One is immediately reduced to proving that filtering $U$-limits commute with fiber products. The proof is left to the reader. One may use the description of the limit given by the following lemma.

Lemma 2.8.1. Let $I$ be a small filtering category and let $i\mapsto X_i$ be a functor from $I$ to $U\text{-}\mathbf{Ens}$. On the sum set ${\coprod }_{i\in \mathrm{ob}I}{X}_i$, let $R$ be the following relation:

(R) Two elements $x_i\in X_i$ and $x_j\in X_j$ are related if there exists an object $k\in \mathrm{ob}I$ and two morphisms $u:i\to k$, $v:j\to k$ such that the images of $x_i$ and $x_j$ by the transition maps $u$ and $v$, respectively, are equal.

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Then:

1. $R$ is an equivalence relation.
2. The quotient $({\coprod }_{i\in \mathrm{ob}I}{X}_i)/R$ is canonically isomorphic to ${\underrightarrow{\lim }}_I{X}_i$.
3. Two equivalent elements are respectively equivalent to two elements of a same $X_i$.
4. For every $i\in \mathrm{ob}I$, two elements $\alpha$ and $\beta$ of $X_i$ are equivalent under $R$ if and only if there exists a morphism $u:i\to j$ such that $u(\alpha )=u(\beta )$.

Assertion 1 follows from PS 1 (2.7). To prove 2, one checks that $(\coprod X_i)/R$ has the universal property of the inductive limit; only PS 1 is used. Assertion 3 follows from the fact that $I$ is connected. Assertion 4 follows from PS 2.

Corollary 2.9. Let $Y$ be a species of algebraic structure “defined by finite projective limits.” The reader is asked to give a mathematical meaning to this phrase; let us merely note that structures of groups, abelian groups, rings, modules, and so on are such structures. Denote by $U\text{-}Y$ the category of $Y$-objects in $U\text{-}\mathbf{Ens}$. The functor which associates to each object of $U\text{-}Y$ its underlying set commutes with filtering $U$-limits. Consequently, filtering $U$-limits in $U\text{-}Y$ commute with finite projective limits.

Corollary 2.10. Pseudo-filtering $U$-limits in $U\text{-}\mathbf{Ab}$ commute with finite projective limits.

One reduces to filtering limits by decomposing the index category into connected components.

Proposition 2.11. Let $C$ and $C^{\prime }$ be two $U$-categories, and let $u:C\to C^{\prime }$, $v:C^{\prime }\to C$ be two functors, with $u$ left adjoint to $v$. Recall that this means that there is an isomorphism of bifunctors with values in $U\text{-}\mathbf{Ens}$

$${\mathrm{Hom}}_{C^{\prime }}(u(X),X^{\prime })\simeq {\mathrm{Hom}}_C(X,v(X^{\prime })).$$

The functor $u$ commutes with representable inductive limits; the functor $v$ commutes with representable projective limits. This assertion means that, for every category $I$ and every functor $G:I\to C^{\prime }$ such that the projective limit of $G$ is representable, the functor $vG$ admits a representable projective limit and one has a canonical isomorphism

$$v(\underset{I}{\underleftarrow{\lim }}G)\simeq \underset{I}{\underleftarrow{\lim }}(vG).$$

The reader will spell out for himself the assertion concerning the functor $u$.

Proposition 2.12. Let $I$ and $C$ be two categories, and let $V$ be a universe. Assume that the $V$-projective limits in $C$ are representable and that there exists in $I$ a family of objects $(i_{\alpha }{)}_{\alpha \in A}$, with $A\in V$, such that:

1. the products $i_{\alpha }\times i_{\beta }$ are representable for every pair $(\alpha ,\beta )\in A\times A$;
2. every object of $I$ maps to at least one of the $i_{\alpha }$.

Then, for every contravariant functor $F:I^{\circ }\to C$, the projective limit of $F$ exists and there is an isomorphism functorial in $F$:

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$$\underleftarrow{\lim }F\simeq \mathrm{Ker}\left(\prod _{\alpha \in A}F(i_{\alpha })\rightrightarrows \prod _{(\alpha ,\beta )\in A\times A}F(i_{\alpha }\times i_{\beta })\right),$$

the two arrows being defined by the projections of the products $i_{\alpha }\times i_{\beta }$ onto the factors.

3. Exactness Properties of the Category of Presheaves

Let $C$ be a $U$-category and let $\hat{C}$ be the category of presheaves on $C$. The exactness properties of $\hat{C}$ all follow from the following proposition.

Proposition 3.1. The $U$-projective and inductive limits in $\hat{C}$ are representable. For every object $X$ of $C$, the functor on $\hat{C}$

$$F\mapsto F(X)$$

commutes with inductive and projective limits. In other words, inductive and projective limits in $\hat{C}$ are computed argument by argument.

Let us state a few corollaries.

Corollary 3.2. Let $Y$ be an algebraic structure defined by finite projective limits. The category of contravariant functors with values in $U\text{-}Y$ is equivalent to the category of $Y$-objects of $\hat{C}$.

Corollary 3.3. A morphism of $\hat{C}$ which is both a monomorphism and an epimorphism is an isomorphism. A morphism of $\hat{C}$ factors uniquely as an epimorphism followed by a monomorphism. Inductive limits in $\hat{C}$ which are representable are universal. Filtering $U$-inductive limits in $\hat{C}$ commute with finite projective limits. The canonical functor

$$C\to \hat{C}$$

commutes with representable projective limits.

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More generally, one can say that the category $\hat{C}$ inherits all the properties of the category of $U$-sets which involve inductive and projective limits.

3.4.0. Let $F$ be an object of $\hat{C}$. We denote by $C/F$ the following category. The objects of $C/F$ are pairs formed by an object $X$ of $C$ and a morphism $u:X\to F$. Let $(X,u)$ and $(Y,v)$ be two objects. A morphism from $(X,u)$ to $(Y,v)$ is a morphism $g:X\to Y$ such that the diagram

X --g--> Y
 \       |
  u      v
   \     |
      F

is commutative.

Proposition 3.4. With the notation of 3.4.0, the source functor

$$C/F\to \hat{C}$$

admits a representable inductive limit in $\hat{C}$. The canonical morphism

$$\underset{C/F}{\underrightarrow{\lim }}\mathrm{source}(-)\to F$$

is an isomorphism.

Corollary 3.5. Let $F$ and $H$ be two objects of $\hat{C}$. There is a canonical isomorphism

$$\mathrm{Hom}(F,H)\simeq \underset{(X,u)\in C/F}{\underleftarrow{\lim }}H(X).$$

Proof. The corollary follows immediately from 3.4 and 1.2.

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Proposition 3.6. Let $C$ be a $U$-category, let $V$ be a universe containing $U$, and let

$$i:{\hat{C}}_U\to {\hat{C}}_V$$

be the natural injection functor of the corresponding categories of presheaves (1.3). The functor $i$ commutes with inductive and projective limits. Moreover, for every object $F$ of ${\hat{C}}_U$, every subobject of $i(F)$ is isomorphic to the image under $i$ of a unique subobject of $F$; thus one obtains a bijection between the set of subobjects of $F$ and the set of subobjects of $i(F)$.

4. Sieves

Definition 4.1. Let $C$ be a category. A sieve of the category $C$ is a full subcategory $D$ of $C$ having the following property: every object of $C$ for which there exists a morphism from this object to an object of $D$ lies in $D$. Let $X$ be an object of $C$; by abuse of language, the sieves of $X$ are the sieves of the category $C/X$.

Let $U$ be a universe such that $C$ is a $U$-category. Let $\hat{C}$ be the corresponding category of presheaves. To every sieve of $X$, one associates a subobject of $X$ in $\hat{C}$ as follows: to every object $Y$ of $C$, one assigns the set of morphisms $f:Y\to X$ such that the object $(Y,f)$ belongs to the sieve.

Proposition 4.2. The map defined above establishes a bijection between the set of sieves of $X$ and the set of subobjects of $X$ in $\hat{C}$.

Proof. We show only that it is the inverse map. To every subfunctor $R$ of $X$, one associates the category $C/R$ of objects of $C$ over $R$ (3.4). It is checked immediately that $C/R$ is a sieve of $X$.

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Remark 4.2.1. Similarly, one sees that the sieves of $C$ are in canonical one-to-one correspondence with the set of subfunctors of the final functor on $C$, the final object of $\hat{C}$.

4.3. Let $C$ be a $U$-category. By abuse of language, we shall also call sieves of $X$ the subobjects of $X$ in the category $\hat{C}$. This abuse of language makes it possible, for every presheaf $F$ and every sieve $R$ of $X$, to define $\mathrm{Hom}(R,F)$ as the set of morphisms from the functor $R$ to $F$. Moreover, there is a canonical isomorphism, functorial in $F$,

$$\mathrm{Hom}(R,F)\simeq \underset{(Y\to X)\in C/R}{\underleftarrow{\lim }}F(Y),$$

which gives a direct definition. Similarly, Proposition 4.2 allows us to transpose to sieves the usual operations on functors. Let us cite:

4.3.1. Base change. Let $R$ be a sieve of $X$ and let $f:Y\to X$ be a morphism of objects of $C$. The fiber product $R{\times }_{X}Y$ is a sieve of $Y$, called the sieve deduced from $R$ by base change. The corresponding subcategory of $C/Y$ is the inverse image of the subcategory of $C/X$ defined by $R$, under the canonical functor $C/Y\to C/X$ defined by $f$.

4.3.2. Order relation; intersection, union. The inclusion relation on subfunctors of $X$ is an order relation. One can define the union and the intersection of a family of sieves indexed by an arbitrary set as the least upper bound and the greatest lower bound of the family of corresponding subpresheaves.

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4.3.3. Image, generated sieve. Let $(F_{\alpha }{)}_{\alpha \in A}$ be a family of presheaves and, for each $\alpha \in A$, let $f_{\alpha }:F_{\alpha }\to X$ be a morphism, where $X$ is an object of $C$. The image of this family of morphisms is the union of the images of the $f_{\alpha }$. The image of this family is therefore a sieve of $X$. In particular, if all the $F_{\alpha }$ are objects of $C$, the image sieve will be called the sieve generated by the morphisms $f_{\alpha }$. The category $C/R$ is the full subcategory of $C/X$ formed by the objects $X^{\prime }\to X$ over $X$ such that there exists an $X$-morphism from $X^{\prime }$ to one of the $F_{\alpha }$.

As an exercise, the reader may translate the relations and operations defined here on subfunctors into terms of the categories $C/R$. He will then find that these relations and operations do not depend on the universe $U$ for which $C$ is a $U$-category, and hence that they are defined for every category without being obliged to specify the universe to which the sets of morphisms belong; this could in any case be foreseen a priori from 3.6.

5. Functoriality of Categories of Presheaves

5.0. Let $C$, $C^{\prime }$, and $D$ be three categories and let $u:C\to C^{\prime }$ be a functor. We shall denote by $u^{\ast }$ the functor

$$u^{\ast }:\mathrm{Hom}((C^{\prime }{)}^{\circ },D)\to \mathrm{Hom}(C^{\circ },D),G\mapsto G\circ u,$$

obtained by composing with the functor $u$. The functor $u^{\ast }$ commutes with inductive and projective limits.

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Proposition 5.1. Suppose that $C$ is small and that, in $D$, the $U$-inductive (resp. projective) limits are representable. The functor $u^{\ast }$ admits a left adjoint $u_{!}$ (resp. a right adjoint $u_{\ast }$). Thus one has isomorphisms

$$\mathrm{Hom}(u_{!}F,G)\simeq \mathrm{Hom}(F,u^{\ast }G),$$

and, respectively,

$$\mathrm{Hom}(u^{\ast }G,F)\simeq \mathrm{Hom}(G,u_{\ast }F).$$

Proof. We shall indicate only the proof of the existence of the left adjoint functor. The “respectively” part of the proposition then follows formally from the isomorphisms

$$\mathrm{Hom}(C^{\circ },D)\simeq \mathrm{Hom}(C,D^{\circ }{)}^{\circ }\simeq \mathrm{Hom}((C^{\circ }{)}^{\circ },D^{\circ }{)}^{\circ }.$$

Let $Y$ be an object of $C^{\prime }$. Denote by $I_Y$ the following category. The objects of $I_Y$ are the pairs $(X,m)$, where $X$ is an object of $C$ and $m:Y\to u(X)$ is a morphism. A morphism from $(X,m)$ to $(X^{\prime },m^{\prime })$ is a morphism $s:X\to X^{\prime }$ such that $m^{\prime }=u(s)m$. Composition of morphisms is defined in the evident way.

Let $f:Y\to Y^{\prime }$ be a morphism of $C^{\prime }$. By composition, the morphism $f$ defines a functor $I_{Y^{\prime }}\to I_Y$. Moreover there is a functor ${\mathrm{Pr}}_Y:I_Y\to C$ which associates to the object $(X,m)$ the object $X$. The corresponding diagram is commutative.

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Now let $F$ be a presheaf on $C$ and put

$$(u_{!}F)(Y)=\underset{I_Y}{\underrightarrow{\lim }}F\circ {\mathrm{Pr}}_Y.\text{\hspace{1em}(5.1.2)}$$

The commutativity of the preceding diagram and the functoriality of the inductive limit make $u_{!}F$ a presheaf on $C^{\prime }$. Let us show that the functor $u_{!}$ is left adjoint to the functor $u^{\ast }$. For this, it suffices to show that, for every presheaf $G$ on $C^{\prime }$, there is a functorial isomorphism

$$\mathrm{Hom}(u_{!}F,G)\simeq \mathrm{Hom}(F,u^{\ast }G).$$

Let $\xi :u_{!}F\to G$. For every object $X$ of $C$, the pair $(X,{\mathrm{id}}_{u(X)})$ is an object of $I_{u(X)}$ and, by definition of the inductive limit, one has a canonical morphism

$$F(X)\to (u_{!}F)(u(X)).$$

One deduces, for every object $X$ of $C$, a morphism

$$F(X)\to G(u(X)),$$

visibly functorial in $X$, hence a morphism $F\to u^{\ast }G$.

Conversely, let $\eta :F\to u^{\ast }G$. For every object $Y$ of $C^{\prime }$, the morphisms

$$F(X)\to G(u(X))\to G(Y),$$

associated with the objects $(X,m:Y\to u(X))$ of $I_Y$, define a morphism from the functor $F\circ {\mathrm{Pr}}_Y$ to the constant functor with value $G(Y)$. Thus one obtains a morphism

$$(u_{!}F)(Y)\to G(Y),$$

functorial in $Y$, and consequently a morphism $u_{!}F\to G$.

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The reader will check that the two maps thus defined are inverse to one another, thereby completing the proof.

Proposition 5.2. Suppose that, in $D$, the $U$-inductive limits are representable, finite projective limits are representable, and filtering $U$-inductive limits commute with finite projective limits. Suppose moreover that, in $C$, finite projective limits are representable and that the functor $u$ is left exact (2.3.2). Then finite projective limits are representable in $\mathrm{Hom}(C^{\circ },D)$ and in $\mathrm{Hom}((C^{\prime }{)}^{\circ },D)$, and the functor $u_{!}$ is left exact.

Proof. The first assertion is trivial. Let us prove the second. From the proof of 5.1, for every presheaf $F$ on $C$ and every object $Y$ of $C^{\prime }$, one has

$$(u_{!}F)(Y)=\underset{I_Y}{\underrightarrow{\lim }}F\circ {\mathrm{Pr}}_Y.$$

It therefore suffices to show that the category $I_Y$ satisfies axioms (PS 1) and (PS 2) (2.7) and that this category is connected. The verification is left to the reader.

5.3. Specializing these results to the case where $D$ is the category of $U$-sets, one obtains a sequence of three functors

$$u_{!},u^{\ast },u_{\ast },$$

which is a “sequence of adjoint functors,” in the sense that, for two consecutive functors in the sequence, the one on the right is right adjoint to the other.

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Their essential properties are summarized in the following proposition.

Proposition 5.4. Let $C$ be a small category, let $C^{\prime }$ be a $U$-category, and let $u:C\to C^{\prime }$ be a functor.

1. The functor $u^{\ast }:\widehat{C^{\prime }}\to \hat{C}$ commutes with inductive and projective limits.

2. The functor $u_{\ast }:\hat{C}\to \widehat{C^{\prime }}$ commutes with projective limits. For every presheaf $F$ on $C$ and every object $Y$ of $C^{\prime }$, one has

   $$(u_{\ast }F)(Y)\simeq {\mathrm{Hom}}_{\hat{C}}(u^{\ast }Y,F).$$

3. The functor $u_{!}:\hat{C}\to \widehat{C^{\prime }}$ commutes with inductive limits. The functor $u_{!}$ is defined only up to isomorphism, but it can always be chosen so that the diagram

   C  --u-->  C'
   |         |
   h         h'
   v         v
   C^ --u_!--> (C')^
   

   is commutative, where $h$ and $h^{\prime }$ are the canonical inclusion functors. For every presheaf $F$ on $C$, one has

   $$u_{!}F\simeq \underset{(X\to F)\in C/F}{\underrightarrow{\lim }}{h}^{\prime }(u(X)).$$

4. If finite projective limits are representable in $C$ and if $u$ is left exact (2.3.2), the functor $u_{!}$ is left exact.

Proof. Assertion (1) is trivial. Assertion (2) follows from the fact that $u_{\ast }$ is a right adjoint functor, by (2.11) and (I.4). The same holds for assertion (3), but one applies (3.4) in addition. Finally, assertion (4) is exactly 5.2, which can be applied thanks to (2.7).

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Proposition 5.5. Let $C$ and $C^{\prime }$ be two small categories and

C  --u-->  C'
C' --v-->  C

a pair of functors, where $v$ is left adjoint to $u$. Then there are isomorphisms, compatible with the adjunction isomorphisms:

$$v^{\ast }\simeq u_{!},v_{\ast }\simeq u^{\ast }.$$

Proof. It suffices to exhibit an isomorphism $v^{\ast }\simeq u_{!}$; the other isomorphism follows from it by adjunction. Let $F$ be a presheaf on $C$ and let $Y$ be an object of $C^{\prime }$. Then

$$v^{\ast }F(Y)\simeq \mathrm{Hom}(v(Y),F).$$

Then, using (3.4),

$$\mathrm{Hom}(v(Y),F)\simeq \underset{C/F}{\underrightarrow{\lim }}\mathrm{Hom}(v(Y),-).$$

But $v$ is left adjoint to $u$, and therefore

$$\underset{C/F}{\underrightarrow{\lim }}\mathrm{Hom}(v(Y),-)\simeq \underset{C/F}{\underrightarrow{\lim }}\mathrm{Hom}(Y,u(-)).$$

Using 5.4 3), it follows that

$$\underset{C/F}{\underrightarrow{\lim }}\mathrm{Hom}(Y,u(-))\simeq \mathrm{Hom}(Y,u_{!}F)\simeq u_{!}F(Y).$$

Thus, for every object $Y$ of $C^{\prime }$, we have determined an isomorphism $v^{\ast }F(Y)\simeq u_{!}F(Y)$, visibly functorial in $Y$ and in $F$.

Corollary 5.5.1. Let $u:C\to C^{\prime }$ be a functor which admits a left adjoint. The functor $u_{!}:\hat{C}\to \widehat{C^{\prime }}$ commutes with projective limits; recall that it commutes with inductive limits by (5.4, 3).

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Remark 5.5.2. Thus one finds a “sequence of four adjoint functors” (cf. 5.3),

$$v_{!},v^{\ast }=u_{!},v_{\ast }=u^{\ast },u_{\ast },$$

of which the first three (resp. last three) therefore commute with inductive (resp. projective) limits.

Proposition 5.6. The hypotheses are those of 5.4. The following conditions are equivalent:

1. The functor $u$ is fully faithful.
2. The functor $u_{!}$ is fully faithful.
3. The adjunction morphism $\mathrm{id}\to u^{\ast }{u}_{!}$ is an isomorphism.
4. The functor $u_{\ast }$ is fully faithful.
5. The adjunction morphism $u^{\ast }{u}_{\ast }\to \mathrm{id}$ is an isomorphism.

Proof. It is clear that (2) is equivalent to (3) and that (4) is equivalent to (5), by the general properties of adjoint functors, and that (2) implies (1) by 5.4 3). Let us show that (1) implies (3). The functors $\mathrm{id}$, $u^{\ast }$, and $u_{!}$ commute with inductive limits. By (3.4), it therefore suffices to prove that $H\to u^{\ast }{u}_{!}H$ is an isomorphism when $H$ is representable, which is evident.

Let us show that (3) is equivalent to (5). For every object $H$ of $\hat{C}$ (resp. $K$ of $\widehat{C^{\prime }}$), denote by $\phi (H):H\to u^{\ast }{u}_{!}H$ (resp. by $\psi (K):u^{\ast }{u}_{\ast }K\to K$) the adjunction morphism. Then one has a commutative diagram:

Hom_{C^}(H, u*u_*K) --Hom(H, psi(K))--> Hom_{C^}(H, K)
        |                                      ^
        ~=                                     |
        v                                      Hom(phi(H), K)
Hom_{(C')^}(u_!H, u_*K) -------------> Hom_{C^}(u*u_!H, K).

It follows that $\phi (H)$ is an isomorphism for every $H$ if and only if $\psi (K)$ is an isomorphism for every $K$.

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Remark 5.7.

1. The equivalences $(2)⟺(3)$ and $(4)⟺(5)$ are general results on adjoint functors.
2. The explicit form of $u_{!}$ and $u_{\ast }$, given in the proof of 5.1, shows at once that, under the general hypotheses of 5.1, if $u$ is fully faithful, then the adjunction morphism $\mathrm{id}\to u^{\ast }{u}_{!}$ (resp. $u^{\ast }{u}_{\ast }\to \mathrm{id}$) is an isomorphism, that is, $u_{!}$ (resp. $u_{\ast }$) is fully faithful.

5.8.0. Let $Y$ be a species of algebraic structure defined by finite projective limits, and let $U\text{-}Y\text{-}\mathbf{Ens}$ be the category of $Y$-objects of $U\text{-}\mathbf{Ens}$. Denote by

$$\mathrm{ens}:U\text{-}Y\text{-}\mathbf{Ens}\to U\text{-}\mathbf{Ens}$$

the “underlying set” functor. For simplicity, we suppose that the species of structure under consideration has a single underlying set. Composition with $\mathrm{ens}$ provides a functor, denoted

$${\mathrm{ens}}^C:\mathrm{Hom}(C^{\circ },U\text{-}Y\text{-}\mathbf{Ens})\to \hat{C}.$$

Since, in $\hat{C}$, projective limits are computed argument by argument, the functor ${\mathrm{ens}}^C$ factors through an equivalence

$$\mathrm{Hom}(C^{\circ },U\text{-}Y\text{-}\mathbf{Ens})\simeq (\hat{C})Y,\text{\hspace{1em}(5.8.1)}$$

where $(\hat{C})Y$ denotes the category of $Y$-objects of $\hat{C}$, and then through a functor still denoted

$${\mathrm{ens}}^C:(\hat{C})Y\to \hat{C},$$

called the “underlying presheaf of sets” functor.

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5.8.2. Suppose that the functor $\mathrm{ens}:U\text{-}Y\text{-}\mathbf{Ens}\to U\text{-}\mathbf{Ens}$ admits a left adjoint

$$\mathrm{Lib}:U\text{-}\mathbf{Ens}\to U\text{-}Y\text{-}\mathbf{Ens}.$$

In fact one can show that this condition is always satisfied. Composition with $\mathrm{Lib}$ provides a functor

$${\mathrm{Lib}}^C:\hat{C}\to \mathrm{Hom}(C^{\circ },U\text{-}Y\text{-}\mathbf{Ens}),$$

and, composing with the equivalence (5.8.1), a functor still denoted

$${\mathrm{Lib}}^C:\hat{C}\to (\hat{C})Y,$$

called the “presheaf of free $Y$-objects generated by” functor. The functor ${\mathrm{Lib}}^C$ is left adjoint to the functor ${\mathrm{ens}}^C$.

Proposition 5.8.3. Let $Y$ be a species of algebraic structure defined by finite projective limits such that, in the category of $Y$-objects of $U\text{-}\mathbf{Ens}$, the $U$-inductive limits are representable; let $C$ be a category belonging to $U$, let $C^{\prime }$ be a $U$-category, and let $u:C\to C^{\prime }$ be a functor. Denote by $(\hat{C})Y$ (resp. $(\widehat{C^{\prime }})Y$) the category of $Y$-objects of $\hat{C}$ (resp. of $\widehat{C^{\prime }}$), and by $u^{\ast }Y$ the functor on $Y$-objects deduced from the functor $u^{\ast }$. It follows from 5.1 and the equivalence (5.8.1) that there exists a functor left adjoint (resp. right adjoint) to the functor $u^{\ast }Y$. This functor is denoted $u_{!}Y$ (resp. $u_{\ast }Y$).

1. The functor $u^{\ast }Y$ commutes with inductive and projective limits. The diagram

   ((C')^)Y --u*Y--> (C^)Y
      |                |
    ens              ens
      v                v
    (C')^  --u*-->    C^
   

   is commutative, where the vertical functors are the “underlying set” functors.

2. The functor $u_{\ast }Y$ commutes with projective limits. The diagram

   (C^)Y --u_*Y--> ((C')^)Y
     |                |
   ens              ens
     v                v
    C^  --u_*-->    (C')^
   

   is commutative up to isomorphism.

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3. The functor $u_{!}Y$ commutes with inductive limits. Suppose that ${\mathrm{ens}}^C$ (resp. ${\mathrm{ens}}^{C^{\prime }}$) admits a left adjoint ${\mathrm{Lib}}^C$ (resp. ${\mathrm{Lib}}^{C^{\prime }}$) as in 5.8.2. The diagram

   C^ --u_!--> (C')^
   |             |
   Lib^C       Lib^{C'}
   v             v
   (C^)Y --u_!Y--> ((C')^)Y
   

   is commutative up to isomorphism.

Finally suppose that $u_{!}$ commutes with finite projective limits (5.4 4 and 5.6). Then the diagram

(C^)Y --u_!Y--> ((C')^)Y
  |                |
ens              ens
  v                v
 C^  --u_!-->    (C')^

is commutative up to isomorphism, and $u_{!}Y$ commutes with finite projective limits.

Proof. Assertion (1) is evident. Assertion (2) is also evident, since $u_{\ast }Y$ is a right adjoint and therefore commutes with projective limits by (2.11), in particular with finite projective limits; whence the commutativity of the corresponding diagram. The commutativity of the diagram in (3) follows from uniqueness, up to isomorphism, of the left adjoint functor. Finally, the commutativity of the last diagram follows immediately from the fact that $u_{!}$ commutes with finite projective limits.

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Notation 5.9. By abuse of notation, the functors $u^{\ast }Y$ and $u_{\ast }Y$ will often be denoted $u^{\ast }$ and $u_{\ast }$, which cannot cause confusion in view of the commutativity of the first two diagrams above. On the other hand, when $u_{!}$ does not commute with finite projective limits, the last diagram is not commutative up to isomorphism, and the notations $u_{!}$ and $u_{!}Y$ must be distinguished to avoid any confusion.

5.10. Let $C$ be a small category. For every object $X$ of $\hat{C}$, denote by $C/X$ the category of arrows with target $X$ and source an object of $C$. The source functor defines a functor

$$j_X:C/X\to C.$$

Let $m:Y\to X$ be a morphism of $\hat{C}$. Composition of morphisms defines a functor

$$j_m:C/Y\to C/X,$$

and the evident diagram with the source functors is commutative. It follows from 5.1 that, for every object $F$ of $(C/X{)}^{\wedge }$ and every object $Y$ of $C$, one has

$$(j_{X\ast }F)(Y)=\prod _{u\in {\mathrm{Hom}}_{\hat{C}}(Y,X)}F(u).\text{\hspace{1em}(5.10.1)}$$

Formula (5.10.1) makes it possible to define $j_{X\ast }$ when $C$ is a $U$-category, and one checks that the functor thus defined is always right adjoint to the functor

$$j_X^{\ast }:\hat{C}\to (C/X{)}^{\wedge }.$$

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Proposition 5.11. Let $C$ be a $U$-category and let $X$ be a presheaf on $C$.

1. The functor

   $$e_X:(C/X{)}^{\wedge }\to \hat{C}$$

   factors through the category $\hat{C}/X$:

   $$(C/X{)}^{\wedge }\stackrel{e_X}{\to }\hat{C}/X\to \hat{C}.$$

   The functor $e_X:(C/X{)}^{\wedge }\to \hat{C}/X$ is an equivalence of categories.

2. The functor $e_X\circ j_X^{\ast }:\hat{C}\to \hat{C}/X$ is canonically isomorphic to the functor

   $$H\mapsto (H\times X\to X).$$

Proof. For (1), let $\epsilon$ be the final object of $(C/X{)}^{\wedge }$. There is a canonical isomorphism

$$\epsilon \simeq \underset{(Y\to X)\in C/X}{\underrightarrow{\lim }}Y,$$

and hence $e_X(\epsilon )\simeq \underrightarrow{\lim }{e}_X(Y)$. But $e_X(\epsilon )=X$, whence the factorization. To show that $e_X$ is an equivalence, it suffices to exhibit a quasi-inverse: to every object $H\to X$ of $\hat{C}/X$ one associates the presheaf on $C/X$

$$(Y\to X)\mapsto {\mathrm{Hom}}_{\hat{C}/X}((Y\to X),(H\to X)).$$

2. The functor $e_X\circ j_X^{\ast }$ is right adjoint to the forgetful functor, and consequently it is canonically isomorphic to the functor $H\mapsto (H\times X\to X)$.

5.12. Let $m:Y\to X$ be a morphism of $\hat{C}$. By 5.11, the morphism $m$ is canonically isomorphic to the image under $e_X$ of an object of $(C/X{)}^{\wedge }$, which we shall denote $[m]$. The functor $e_X$ defines, by restriction to subcategories, an equivalence

$$(C/X)/[m]\simeq C/Y.$$

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The evident diagram is commutative up to canonical isomorphism.

5.13. Let us record a result which will be useful in Expose VI. Let $u:C\to C^{\prime }$ be a functor between small categories. For every object $H$ of $\hat{C}$, denote by

$$u/H:C/H\to C^{\prime }/u_{!}H$$

the functor which associates to every morphism $m:X\to H$ the morphism $u_{!}X\to u_{!}H$. We know by 5.4 that one can always put $u_{!}X=uX$. One obtains a commutative diagram

C/H --u/H--> C'/u_!H
 |             |
 j_H           j_{u_!H}
 v             v
 C  --u-->     C'.                         (5.13.1)

Thus one has a diagram commutative up to isomorphism

(C/H)^ ----> (C'/u_!H)^
  |              |
 j_{H*}        j_{u_!H,*}
  v              v
 C^ --u_*-->   (C')^.                      (5.13.2)

Since $(u/H{)}_{!}$ transforms the final object of $(C/H{)}^{\wedge }$ into the final object of $(C^{\prime }/u_{!}H{)}^{\wedge }$, the diagram

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(C/H)^ ----> (C'/u_!H)^
  |              |
 e_H          e_{u_!H}
  v              v
 C^/H ----> (C')^/u_!H                    (5.13.3)

is commutative up to isomorphism.

Proposition 5.14. Let $u:C\to C^{\prime }$ be a functor between small categories. Assume that $u$ has the following property:

(PPF) For every object $X$ of $C$, the functor

$$u/X:C/X\to C^{\prime }/uX$$

is fully faithful.

Then:

1. Let $\epsilon$ be the final object of $\hat{C}$. The functor $u$ factors as

   $$C\simeq C/\epsilon \stackrel{u/\epsilon }{\to }{C}^{\prime }/u_{!}\epsilon \to C^{\prime }.$$

   The functor $u/\epsilon$ is fully faithful.

2. The functor $u_{!}:\hat{C}\to \widehat{C^{\prime }}$ factors as

   $$\hat{C}\simeq (C/\epsilon {)}^{\wedge }\stackrel{(u/\epsilon {)}_{!}}{\to }(C^{\prime }/u_{!}\epsilon {)}^{\wedge }\stackrel{e_{u_{!}\epsilon }}{\to }\widehat{C^{\prime }}/u_{!}\epsilon \to \widehat{C^{\prime }},$$

   where the functor $(u/\epsilon {)}_{!}$ is fully faithful, the functor $e_{u_{!}\epsilon }$ is an equivalence, and the last functor is the forgetful functor.

3. In particular, the functor $u_{!}$ is faithful; consequently the adjunction morphism

   $$\mathrm{id}\to u^{\ast }{u}_{!}$$

   is a monomorphism. Moreover, for every morphism $\alpha :H\to K$ of $\hat{C}$, the square

   H      ->  u*u_!H
   |          |
   alpha      u*u_!(alpha)
   v          v
   K      ->  u*u_!K
   

   is cartesian.

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Proof. 1. The factorization comes from diagram (5.13.1). The functor $u$ is faithful; hence $u/\epsilon$ is faithful. Let us show that it is fully faithful. Let $X$ and $Y$ be two objects of $C$, let ${\mathrm{can}}_X:uX\to u_{!}\epsilon$ and ${\mathrm{can}}_Y:uY\to u_{!}\epsilon$ be the canonical morphisms, and let $m:uX\to uY$ be a morphism of $C^{\prime }/u_{!}\epsilon$. One has

$$u_{!}\epsilon =\underset{Z\in C}{\underrightarrow{\lim }}uZ.$$

By definition of the inductive limit, saying that ${\mathrm{can}}_{Y}m={\mathrm{can}}_X$ is equivalent to saying that there exists a finite sequence of objects $X_i$ of $C$, with $X_0=X$ and $X_n=Y$, and, for every pair $(i,i+1)$, a morphism $X_i\to X_{i+1}$ or a morphism $X_{i+1}\to X_i$, so that the corresponding diagrams are commutative. One then proves immediately, by induction on $i$ and using property (PPF), that $m$ is of the form $u(p)$. In particular $u/\epsilon$ is fully faithful.

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2. The factorization is immediate. The functor $(u/\epsilon {)}_{!}$ is fully faithful by 5.6. The other assertions follow from 5.11.

3. The functor $u_{!}$ is the composite of the forgetful functor, which is faithful, and fully faithful functors; it is therefore faithful. It follows, from the general properties of adjoint functors, that the adjunction morphism $\mathrm{id}\to u^{\ast }{u}_{!}$ is a monomorphism. By (2), the functor $u_{!}$ appears as the composite of a fully faithful functor $v:\hat{C}\to \widehat{C^{\prime }}/u_{!}\epsilon$ and the forgetful functor. The functor $u^{\ast }$, right adjoint to $u_{!}$, is therefore the composite of the “product by $u_{!}\epsilon$” functor, right adjoint to the forgetful functor, and of a functor $w$ right adjoint to $v$. Moreover, since $v$ is fully faithful, the adjunction morphism $\mathrm{id}\to wv$ is an isomorphism. The last assertion follows easily.

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6. Faithful Functors and Conservative Functors

Definition 6.1. Let $E$ be a category and let $({\phi }_i:E\to F_i{)}_{i\in I}$ be a family of functors. We say that the family $({\phi }_i)$ is faithful if, for every pair of objects $X,Y$ of $E$ and every pair of arrows $u,v:X\to Y$, the relation ${\phi }_i(u)={\phi }_i(v)$ for every $i\in I$ implies $u=v$. In other words, the map

$${\mathrm{Hom}}_E(X,Y)\to \prod _{i\in I}{\mathrm{Hom}}_{F_i}({\phi }_{i}X,{\phi }_{i}Y)$$

defined by the ${\phi }_i$ is injective.

We say that the family $({\phi }_i)$ is conservative if every arrow $u$ of $E$ such that ${\phi }_i(u)$ is an isomorphism for every $i\in I$ is an isomorphism. We say that $({\phi }_i)$ is conservative for monomorphisms (resp. for epimorphisms, etc.) if the preceding condition is verified whenever $u$ is a monomorphism (resp. an epimorphism, etc.).

6.1.1. If one introduces the unique functor

$$\phi :E\to F=\prod _{i\in I}{F}_i$$

defined by the family $({\phi }_i)$, it is clear that this family is faithful (resp. conservative, resp. conservative for monomorphisms, etc.) if and only if the functor $\phi$ is faithful (resp. conservative, resp. etc.). Thus, without serious inconvenience, we could restrict ourselves below to the case of a family reduced to a single functor. For convenience in later references, nevertheless, we shall give the following statements for families.

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The notions of 6.1 are especially useful when the ${\phi }_i$ satisfy suitable exactness properties; in that case, they tend to coincide.

Proposition 6.2. The notation is that of 6.1.

1. If kernels of double arrows, or cokernels of double arrows, are representable in $E$, and if the ${\phi }_i$ commute with them, then one has the implication

   $$({\phi }_i)\text{ conservative }⟹({\phi }_i)\text{ faithful}.$$

2. Suppose that fiber products (resp. amalgamated sums) are representable in $E$, and that the ${\phi }_i$ commute with them. Suppose that $({\phi }_i)$ is faithful or conservative. Then, for every arrow $u$ of $E$, $u$ is a monomorphism (resp. an epimorphism) if and only if ${\phi }_i(u)$ is so for every $i\in I$.

3. Suppose that, in $E$, fiber products and amalgamated sums are representable, that the ${\phi }_i$ commute with them, and that every arrow of $E$ which is a bimorphism is an isomorphism. Then one has the implication

   $$({\phi }_i)\text{ faithful }⟹({\phi }_i)\text{ conservative}.$$

4. Suppose that, in $E$, fiber products (resp. amalgamated sums) are representable, and that the ${\phi }_i$ commute with them. If $({\phi }_i)$ is conservative for monomorphisms (resp. for epimorphisms), then $({\phi }_i)$ is conservative.

5. Let $D$ be a type of diagram, let $F:D\to E$ be a diagram of type $D$ in $E$, let $X$ be an object of $E$, and let $u=(u_d{)}_{d\in D}$ be a family of arrows

   $$X\to F(d)(\text{resp. }F(d)\to X).$$

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Suppose that $({\phi }_i)$ is conservative, that projective (resp. inductive) limits of type $D$ are representable in $E$, and that the ${\phi }_i$ commute with them. Then, for $u$ to make $X$ a projective (resp. inductive) limit of $F$ in $E$, it is necessary and sufficient that, for every $i\in I$, ${\phi }_i(u)$ make ${\phi }_i(X)$ a projective (resp. inductive) limit of ${\phi }_i\circ F$ in $F_i$.

Proof. For (1), in the non-dual statement, for a given double arrow $u,v:X\rightrightarrows Y$, it suffices to express the equality $u=v$ by the condition that the inclusion $\mathrm{Ker}(u,v)\to X$ be an isomorphism. Here and below, we omit repeating the dual argument.

For (2), if $({\phi }_i)$ is faithful, one expresses the condition that $u:X\to Y$ be a monomorphism by the equality of the two projections in the fiber product $X{\times }_{Y}X$. If $({\phi }_i)$ is conservative, one expresses it by the condition that the diagonal morphism $\delta :X\to X{\times }_{Y}X$ be an isomorphism.

For (4), as in this last argument, the morphism $\delta$ is a monomorphism, so one sees that it would in fact have sufficed to suppose $({\phi }_i)$ conservative for monomorphisms. But this then implies that $({\phi }_i)$ is conservative outright. Indeed, if $u$ is an arrow of $E$ such that the ${\phi }_i(u)$ are isomorphisms, one first concludes that $u$ is a monomorphism by what precedes, and then an isomorphism by the hypothesis on $({\phi }_i)$.

Assertion (3) is a trivial consequence of (2), and (5) is a trivial consequence of the definitions.

Let us note the following consequence of (1), (2), and (4).

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Corollary 6.3. Suppose that, in $E$, fiber products and amalgamated sums are representable and that the ${\phi }_i$ commute with them; suppose furthermore that kernels of double arrows or cokernels of double arrows are representable and that the ${\phi }_i$ commute with them. It suffices, for example, that finite projective limits and finite inductive limits be representable in $E$, and that the ${\phi }_i$ be exact functors. Then the following conditions are equivalent:

1. $({\phi }_i)$ is faithful.
2. $({\phi }_i)$ is conservative.
3. $({\phi }_i)$ is conservative for monomorphisms.
4. $({\phi }_i)$ is conservative for epimorphisms.

Let us also record for reference:

Proposition 6.4. Let $\phi :E\to F$ be a functor admitting a right adjoint $\psi$, so that

$${\mathrm{Hom}}_F(\phi (X),Y)\simeq {\mathrm{Hom}}_E(X,\psi (Y)).$$

For $\phi$ (resp. $\psi$) to be faithful, it is necessary and sufficient that, for every object $X$ of $E$ (resp. every object $Y$ of $F$), the adjunction morphism $X\to \psi \phi (X)$ (resp. $\phi \psi (Y)\to Y$) be a monomorphism. For $\phi$ (resp. $\psi$) to be fully faithful, it is necessary and sufficient that the corresponding adjunction morphism be an isomorphism.

Indeed, if $X^{\prime }$ and $X$ are two objects of $E$, the map

$${\mathrm{Hom}}_E(X^{\prime },X)\to {\mathrm{Hom}}_F(\phi (X^{\prime }),\phi (X))$$

is identified with the map deduced from the adjunction morphism

$$X\to \psi \phi (X)$$

by applying the functor ${\mathrm{Hom}}_E(X^{\prime },-)$. Thus, for this map to be a monomorphism (resp. an isomorphism) for every $X^{\prime }$, with $X$ fixed, it is necessary and sufficient that the adjunction morphism be a monomorphism (resp. an isomorphism).

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Proposition 6.5. Let $p:E^{\prime }\to E$ be a functor. The following conditions are equivalent:

1. $p$ is faithful, conservative, and fibering (SGA 1 VI 6.1).

2. $p$ is a fibering functor with fibers discrete categories.

3. For every $X^{\prime }\in \mathrm{ob}(E^{\prime })$, the functor on arrow categories induced by $p$

   $$E^{\prime }/X^{\prime }\to E/p(X^{\prime })$$

   is an equivalence of categories, surjective on objects.

4. When $E$ is a $U$-category, there exists a presheaf $F\in \mathrm{ob}(\hat{E})$ and an equivalence of categories over $E$

   $$E^{\prime }\simeq E/F,$$

   where $E/F$ is the full subcategory of $\hat{E}/F$ formed by the arrows $X\to F$ whose source is in $E$.

6.5.1. Recall that a category $C$ is called discrete if it is a groupoid, that is, if every arrow in it is invertible, and if it is rigid, that is, if the automorphism group of every object is reduced to the unit group. This is equivalent to saying that the category is equivalent to the category defined by a set, with only identity arrows. When one already assumes that $C$ is a groupoid, saying that $C$ is discrete amounts to saying that, for two objects $X,Y$ of $C$, there exists at most one arrow from $X$ to $Y$.

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The equivalence of conditions (1) and (2) of 6.5 is an immediate consequence of the preceding reminders and of the following lemma.

Lemma 6.5.2. Let $p:E^{\prime }\to E$ be a fibering functor. Then:

1. For $p$ to be conservative, it is necessary and sufficient that its fiber categories be groupoids.
2. For $p$ to be faithful, it is necessary and sufficient that its fiber categories be ordered categories.

Proof. First suppose that $p$ is conservative. For every arrow $u$ of a fiber $E_X^{\prime }$, $p(u)={\mathrm{id}}_X$ is an isomorphism, hence $u$ is an isomorphism in $E^{\prime }$, and therefore also in $E_X^{\prime }$. Thus $E_X^{\prime }$ is a groupoid. Conversely, suppose that the $E_X^{\prime }$ are groupoids, and let $u^{\prime }$ be an arrow of $E^{\prime }$ such that $p(u^{\prime })$ is an isomorphism. Write $X=p(X^{\prime })$ and $u=p(u^{\prime })$. Since $p$ is fibering, one can factor $u^{\prime }:X^{\prime }\to Y^{\prime }$ as a composite