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*}

4.3 Opposite Categories and the Yoneda Lemma

Definition 4.3.1. Given a category $\mathcal{C}$ the opposite category $\mathcal{C}^{opp}$ is the category with the same objects as $\mathcal{C}$ but all morphisms reversed.

In other words $\mathop{Mor}\nolimits _{\mathcal{C}^{opp}}(x, y) = \mathop{Mor}\nolimits _\mathcal {C}(y, x)$. Composition in $\mathcal{C}^{opp}$ is the same as in $\mathcal{C}$ except backwards: if $\phi : y \to z$ and $\psi : x \to y$ are morphisms in $\mathcal{C}^{opp}$, in other words arrows $z \to y$ and $y \to x$ in $\mathcal{C}$, then $\phi \circ ^{opp} \psi $ is the morphism $x \to z$ of $\mathcal{C}^{opp}$ which corresponds to the composition $z \to y \to x$ in $\mathcal{C}$.

Definition 4.3.2. Let $\mathcal{C}$, $\mathcal{S}$ be categories. A contravariant functor $F$ from $\mathcal{C}$ to $\mathcal{S}$ is a functor $\mathcal{C}^{opp}\to \mathcal{S}$.

Concretely, a contravariant functor $F$ is given by a map $F : \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ and for every morphism $\psi : x \to y$ in $\mathcal{C}$ a morphism $F(\psi ) : F(y) \to F(x)$. These should satisfy the property that, given another morphism $\phi : y \to z$, we have $F(\phi \circ \psi ) = F(\psi ) \circ F(\phi )$ as morphisms $F(z) \to F(x)$. (Note the reverse of order.)

Definition 4.3.3. Let $\mathcal{C}$ be a category.

  1. A presheaf of sets on $\mathcal{C}$ or simply a presheaf is a contravariant functor $F$ from $\mathcal{C}$ to $\textit{Sets}$.

  2. The category of presheaves is denoted $\textit{PSh}(\mathcal{C})$.

Of course the category of presheaves is a proper class.

Example 4.3.4. Functor of points. For any $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there is a contravariant functor

\[ \begin{matrix} h_ U & : & \mathcal{C} & \longrightarrow & \textit{Sets} \\ & & X & \longmapsto & \mathop{Mor}\nolimits _\mathcal {C}(X, U) \end{matrix} \]

which takes an object $X$ to the set $\mathop{Mor}\nolimits _\mathcal {C}(X, U)$. In other words $h_ U$ is a presheaf. Given a morphism $f : X\to Y$ the corresponding map $h_ U(f) : \mathop{Mor}\nolimits _\mathcal {C}(Y, U)\to \mathop{Mor}\nolimits _\mathcal {C}(X, U)$ takes $\phi $ to $\phi \circ f$. We will always denote this presheaf $h_ U : \mathcal{C}^{opp} \to \textit{Sets}$. It is called the representable presheaf associated to $U$. If $\mathcal{C}$ is the category of schemes this functor is sometimes referred to as the functor of points of $U$.

Note that given a morphism $\phi : U \to V$ in $\mathcal{C}$ we get a corresponding natural transformation of functors $h(\phi ) : h_ U \to h_ V$ defined simply by composing with the morphism $U \to V$. It is trivial to see that this turns composition of morphisms in $\mathcal{C}$ into composition of transformations of functors. In other words we get a functor

\[ h : \mathcal{C} \longrightarrow \text{Fun}(\mathcal{C}^{opp}, \textit{Sets}) = \textit{PSh}(\mathcal{C}) \]

Note that the target is a “big” category, see Remark 4.2.2. On the other hand, $h$ is an actual mathematical object (i.e. a set), compare Remark 4.2.11.

reference

Lemma 4.3.5 (Yoneda lemma). Let $U, V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Given any morphism of functors $s : h_ U \to h_ V$ there is a unique morphism $\phi : U \to V$ such that $h(\phi ) = s$. In other words the functor $h$ is fully faithful. More generally, given any contravariant functor $F$ and any object $U$ of $\mathcal{C}$ we have a natural bijection

\[ \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, F) \longrightarrow F(U), \quad s \longmapsto s_ U(\text{id}_ U). \]

Proof. For the first statement, just take $\phi = s_ U(\text{id}_ U) \in \mathop{Mor}\nolimits _\mathcal {C}(U, V)$. For the second statement, given $\xi \in F(U)$ define $s$ by $s_ V : h_ U(V) \to F(V)$ by sending the element $f : V \to U$ of $h_ U(V) = \mathop{Mor}\nolimits _\mathcal {C}(V, U)$ to $F(f)(\xi )$. $\square$

Definition 4.3.6. A contravariant functor $F : \mathcal{C}\to \textit{Sets}$ is said to be representable if it is isomorphic to the functor of points $h_ U$ for some object $U$ of $\mathcal{C}$.

Let $\mathcal{C}$ be a category and let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a representable functor. Choose an object $U$ of $\mathcal{C}$ and an isomorphism $s : h_ U \to F$. The Yoneda lemma guarantees that the pair $(U, s)$ is unique up to unique isomorphism. The object $U$ is called an object representing $F$. By the Yoneda lemma the transformation $s$ corresponds to a unique element $\xi \in F(U)$. This element is called the universal object. It has the property that for $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the map

\[ \mathop{Mor}\nolimits _\mathcal {C}(V, U) \longrightarrow F(V),\quad (f : V \to U) \longmapsto F(f)(\xi ) \]

is a bijection. Thus $\xi $ is universal in the sense that every element of $F(V)$ is equal to the image of $\xi $ via $F(f)$ for a unique morphism $f : V \to U$ in $\mathcal{C}$.


Comments (6)

Comment #1325 by jojo on

This is real nipticking and maybe should be ignored.

The second phrase after Definition 4.3.1 i.e. "Composition in C^{opp}..." is (in my opinion) rather confusing, especially the last equality (indeed isn't defined). It might be better to either say that composition in the opposite category is done in the obvious way or to define things a little bit more carefully.

Comment #1347 by on

Nitpicky indeed! Not sure you are going to like my attempt at improvement which you can find here.

Comment #2149 by Maik Pickl on

Just want to report a very minor typo. In the second to last line it reads "Thus is universal in the sens...", there sens should be sense.

Comment #2377 by Anoop Singh on

In the proof of the Lemma 4.3.5 (Yoneda Lemma ), the last line it is written that but this is a typo i guess, it should be .

Comment #2380 by on

@#2377 Thanks! This was already pointed out by somebody over email and was fixed here.


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