The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

4.2 Definitions

We recall the definitions, partly to fix notation.

Definition 4.2.1. A category $\mathcal{C}$ consists of the following data:

  1. A set of objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

  2. For each pair $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a set of morphisms $\mathop{Mor}\nolimits _\mathcal {C}(x, y)$.

  3. For each triple $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a composition map $ \mathop{Mor}\nolimits _\mathcal {C}(y, z) \times \mathop{Mor}\nolimits _\mathcal {C}(x, y) \to \mathop{Mor}\nolimits _\mathcal {C}(x, z) $, denoted $(\phi , \psi ) \mapsto \phi \circ \psi $.

These data are to satisfy the following rules:

  1. For every element $x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a morphism $\text{id}_ x\in \mathop{Mor}\nolimits _\mathcal {C}(x, x)$ such that $\text{id}_ x \circ \phi = \phi $ and $\psi \circ \text{id}_ x = \psi $ whenever these compositions make sense.

  2. Composition is associative, i.e., $(\phi \circ \psi ) \circ \chi = \phi \circ ( \psi \circ \chi )$ whenever these compositions make sense.

It is customary to require all the morphism sets $\mathop{Mor}\nolimits _\mathcal {C}(x, y)$ to be disjoint. In this way a morphism $\phi : x \to y$ has a unique source $x$ and a unique target $y$. This is not strictly necessary, although care has to be taken in formulating condition (2) above if it is not the case. It is convenient and we will often assume this is the case. In this case we say that $\phi $ and $\psi $ are composable if the source of $\phi $ is equal to the target of $\psi $, in which case $\phi \circ \psi $ is defined. An equivalent definition would be to define a category as a quintuple $(\text{Ob}, \text{Arrows}, s, t, \circ )$ consisting of a set of objects, a set of morphisms (arrows), source, target and composition subject to a long list of axioms. We will occasionally use this point of view.

Remark 4.2.2. Big categories. In some texts a category is allowed to have a proper class of objects. We will allow this as well in these notes but only in the following list of cases (to be updated as we go along). In particular, when we say: “Let $\mathcal{C}$ be a category” then it is understood that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is a set.

  1. The category $\textit{Sets}$ of sets.

  2. The category $\textit{Ab}$ of abelian groups.

  3. The category $\textit{Groups}$ of groups.

  4. Given a group $G$ the category $G\textit{-Sets}$ of sets with a left $G$-action.

  5. Given a ring $R$ the category $\text{Mod}_ R$ of $R$-modules.

  6. Given a field $k$ the category of vector spaces over $k$.

  7. The category of rings.

  8. The category of schemes.

  9. The category $\textit{Top}$ of topological spaces.

  10. Given a topological space $X$ the category $\textit{PSh}(X)$ of presheaves of sets over $X$.

  11. Given a topological space $X$ the category $\mathop{\mathit{Sh}}\nolimits (X)$ of sheaves of sets over $X$.

  12. Given a topological space $X$ the category $\textit{PAb}(X)$ of presheaves of abelian groups over $X$.

  13. Given a topological space $X$ the category $\textit{Ab}(X)$ of sheaves of abelian groups over $X$.

  14. Given a small category $\mathcal{C}$ the category of functors from $\mathcal{C}$ to $\textit{Sets}$.

  15. Given a category $\mathcal{C}$ the category of presheaves of sets over $\mathcal{C}$.

  16. Given a site $\mathcal{C}$ the category of sheaves of sets over $\mathcal{C}$.

One of the reason to enumerate these here is to try and avoid working with something like the “collection” of “big” categories which would be like working with the collection of all classes which I think definitively is a meta-mathematical object.

Remark 4.2.3. It follows directly from the definition that any two identity morphisms of an object $x$ of $\mathcal{A}$ are the same. Thus we may and will speak of the identity morphism $\text{id}_ x$ of $x$.

Definition 4.2.4. A morphism $\phi : x \to y$ is an isomorphism of the category $\mathcal{C}$ if there exists a morphism $\psi : y \to x$ such that $\phi \circ \psi = \text{id}_ y$ and $\psi \circ \phi = \text{id}_ x$.

An isomorphism $\phi $ is also sometimes called an invertible morphism, and the morphism $\psi $ of the definition is called the inverse and denoted $\phi ^{-1}$. It is unique if it exists. Note that given an object $x$ of a category $\mathcal{A}$ the set of invertible elements $\text{Aut}_\mathcal {A}(x)$ of $\mathop{Mor}\nolimits _\mathcal {A}(x, x)$ forms a group under composition. This group is called the automorphism group of $x$ in $\mathcal{A}$.

Example 4.2.6. A group $G$ gives rise to a groupoid with a single object $x$ and morphisms $\mathop{Mor}\nolimits (x, x) = G$, with the composition rule given by the group law in $G$. Every groupoid with a single object is of this form.

Example 4.2.7. A set $C$ gives rise to a groupoid $\mathcal{C}$ defined as follows: As objects we take $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) := C$ and for morphisms we take $\mathop{Mor}\nolimits (x, y)$ empty if $x\neq y$ and equal to $\{ \text{id}_ x\} $ if $x = y$.

Definition 4.2.8. A functor $F : \mathcal{A} \to \mathcal{B}$ between two categories $\mathcal{A}, \mathcal{B}$ is given by the following data:

  1. A map $F : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

  2. For every $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ a map $F : \mathop{Mor}\nolimits _\mathcal {A}(x, y) \to \mathop{Mor}\nolimits _\mathcal {B}(F(x), F(y))$, denoted $\phi \mapsto F(\phi )$.

These data should be compatible with composition and identity morphisms in the following manner: $F(\phi \circ \psi ) = F(\phi ) \circ F(\psi )$ for a composable pair $(\phi , \psi )$ of morphisms of $\mathcal{A}$ and $F(\text{id}_ x) = \text{id}_{F(x)}$.

Note that every category $\mathcal{A}$ has an identity functor $\text{id}_\mathcal {A}$. In addition, given a functor $G : \mathcal{B} \to \mathcal{C}$ and a functor $F : \mathcal{A} \to \mathcal{B}$ there is a composition functor $G \circ F : \mathcal{A} \to \mathcal{C}$ defined in an obvious manner.

Definition 4.2.9. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor.

  1. We say $F$ is faithful if for any objects $x, y$ of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the map

    \[ F : \mathop{Mor}\nolimits _\mathcal {A}(x, y) \to \mathop{Mor}\nolimits _\mathcal {B}(F(x), F(y)) \]

    is injective.

  2. If these maps are all bijective then $F$ is called fully faithful.

  3. The functor $F$ is called essentially surjective if for any object $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ there exists an object $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in $\mathcal{B}$.

Definition 4.2.10. A subcategory of a category $\mathcal{B}$ is a category $\mathcal{A}$ whose objects and arrows form subsets of the objects and arrows of $\mathcal{B}$ and such that source, target and composition in $\mathcal{A}$ agree with those of $\mathcal{B}$. We say $\mathcal{A}$ is a full subcategory of $\mathcal{B}$ if $\mathop{Mor}\nolimits _\mathcal {A}(x, y) = \mathop{Mor}\nolimits _\mathcal {B}(x, y)$ for all $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. We say $\mathcal{A}$ is a strictly full subcategory of $\mathcal{B}$ if it is a full subcategory and given $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ any object of $\mathcal{B}$ which is isomorphic to $x$ is also in $\mathcal{A}$.

If $\mathcal{A} \subset \mathcal{B}$ is a subcategory then the identity map is a functor from $\mathcal{A}$ to $\mathcal{B}$. Furthermore a subcategory $\mathcal{A} \subset \mathcal{B}$ is full if and only if the inclusion functor is fully faithful. Note that given a category $\mathcal{B}$ the set of full subcategories of $\mathcal{B}$ is the same as the set of subsets of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

Remark 4.2.11. Suppose that $\mathcal{A}$ is a category. A functor $F$ from $\mathcal{A}$ to $\textit{Sets}$ is a mathematical object (i.e., it is a set not a class or a formula of set theory, see Sets, Section 3.2) even though the category of sets is “big”. Namely, the range of $F$ on objects will be a set $F(\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}))$ and then we may think of $F$ as a functor between $\mathcal{A}$ and the full subcategory of the category of sets whose objects are elements of $F(\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}))$.

Example 4.2.12. A homomorphism $p : G\to H$ of groups gives rise to a functor between the associated groupoids in Example 4.2.6. It is faithful (resp. fully faithful) if and only if $p$ is injective (resp. an isomorphism).

Example 4.2.13. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the category of objects over $X$, denoted $\mathcal{C}/X$ as follows. The objects of $\mathcal{C}/X$ are morphisms $Y\to X$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $Y\to X$ and $Y'\to X$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : \mathcal{C}/X\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : \mathcal{C}/X' \to \mathcal{C}/X$ obtained by composition with $f$, and $p_ X\circ F = p_{X'}$.

Example 4.2.14. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the category of objects under $X$, denoted $X/\mathcal{C}$ as follows. The objects of $X/\mathcal{C}$ are morphisms $X\to Y$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $X\to Y$ and $X\to Y'$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : X/\mathcal{C}\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : X/\mathcal{C} \to X'/\mathcal{C}$ obtained by composition with $f$, and $p_{X'}\circ F = p_ X$.

Definition 4.2.15. Let $F, G : \mathcal{A} \to \mathcal{B}$ be functors. A natural transformation, or a morphism of functors $t : F \to G$, is a collection $\{ t_ x\} _{x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})}$ such that

  1. $t_ x : F(x) \to G(x)$ is a morphism in the category $\mathcal{B}$, and

  2. for every morphism $\phi : x \to y$ of $\mathcal{A}$ the following diagram is commutative

    \[ \xymatrix{ F(x) \ar[r]^{t_ x} \ar[d]_{F(\phi )} & G(x) \ar[d]^{G(\phi )} \\ F(y) \ar[r]^{t_ y} & G(y) } \]

Sometimes we use the diagram

\[ \xymatrix{ \mathcal{A} \rtwocell ^ F_ G{t} & \mathcal{B} } \]

to indicate that $t$ is a morphism from $F$ to $G$.

Note that every functor $F$ comes with the identity transformation $\text{id}_ F : F \to F$. In addition, given a morphism of functors $t : F \to G$ and a morphism of functors $s : E \to F$ then the composition $t \circ s$ is defined by the rule

\[ (t \circ s)_ x = t_ x \circ s_ x : E(x) \to G(x) \]

for $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. It is easy to verify that this is indeed a morphism of functors from $E$ to $G$. In this way, given categories $\mathcal{A}$ and $\mathcal{B}$ we obtain a new category, namely the category of functors between $\mathcal{A}$ and $\mathcal{B}$.

Remark 4.2.16. This is one instance where the same thing does not hold if $\mathcal{A}$ is a “big” category. For example consider functors $\textit{Sets} \to \textit{Sets}$. As we have currently defined it such a functor is a class and not a set. In other words, it is given by a formula in set theory (with some variables equal to specified sets)! It is not a good idea to try to consider all possible formulae of set theory as part of the definition of a mathematical object. The same problem presents itself when considering sheaves on the category of schemes for example. We will come back to this point later.

Definition 4.2.17. An equivalence of categories $F : \mathcal{A} \to \mathcal{B}$ is a functor such that there exists a functor $G : \mathcal{B} \to \mathcal{A}$ such that the compositions $F \circ G$ and $G \circ F$ are isomorphic to the identity functors $\text{id}_\mathcal {B}$, respectively $\text{id}_\mathcal {A}$. In this case we say that $G$ is a quasi-inverse to $F$.

Lemma 4.2.18. Let $F : \mathcal{A} \to \mathcal{B}$ be a fully faithful functor. Suppose for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ given an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_ X : X \to F(j(X))$. Then there is a unique functor $j : \mathcal{B} \to \mathcal{A}$ such that $j$ extends the rule on objects, and the isomorphisms $i_ X$ define an isomorphism of functors $\text{id}_\mathcal {B} \to F \circ j$. Moreover, $j$ and $F$ are quasi-inverse equivalences of categories.

Proof. This lemma proves itself. $\square$

Lemma 4.2.19. A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective.

Proof. Let $F : \mathcal{A} \to \mathcal{B}$ be essentially surjective and fully faithful. As by convention all categories are small and as $F$ is essentially surjective we can, using the axiom of choice, choose for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_ X : X \to F(j(X))$. Then we apply Lemma 4.2.18 using that $F$ is fully faithful. $\square$

Definition 4.2.20. Let $\mathcal{A}$, $\mathcal{B}$ be categories. We define the product category $\mathcal{A} \times \mathcal{B}$ to be the category with objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A} \times \mathcal{B}) = \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and

\[ \mathop{Mor}\nolimits _{\mathcal{A} \times \mathcal{B}}((x, y), (x', y')) := \mathop{Mor}\nolimits _\mathcal {A}(x, x')\times \mathop{Mor}\nolimits _\mathcal {B}(y, y'). \]

Composition is defined componentwise.


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