Section 4.3 (001L): Opposite Categories and the Yoneda Lemma

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 $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}^{opp}}(x, y) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, x)$. 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.

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

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

which takes an object $X$ to the set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U)$. In other words $h_ U$ is a presheaf. Given a morphism $f : X\to Y$ the corresponding map $h_ U(f) : \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(Y, U)\to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U)$ 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 by composing with the morphism $U \to V$. This turns composition of morphisms in $\mathcal{C}$ into composition of transformations of functors. In other words we get a functor

\[ h : \mathcal{C} \longrightarrow \textit{PSh}(\mathcal{C}) \]

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

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

Proof. For the first statement, just take $\phi = s_ U(\text{id}_ U) \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, V)$. 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 $h_ U(V) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)$ 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

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

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 (12)

Comment #1325 by jojo on February 21, 2015 at 18:33

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 $\psi \circ \phi$ 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 Johan on March 12, 2015 at 19:23

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

Comment #2149 by Maik Pickl on July 25, 2016 at 08:23

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

Comment #2185 by Johan on August 29, 2016 at 14:38

@#2149: Thanks. Fixed here.

Comment #2377 by Anoop Singh on February 14, 2017 at 07:23

In the proof of the Lemma 4.3.5 (Yoneda Lemma ), the last line it is written that $f \in h_V(U)$ but this is a typo i guess, it should be $f \in h_U(V) = Mor_\mathcal{C}(V, U)$ .

Comment #2380 by Johan on February 16, 2017 at 19:58

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

Comment #4960 by Siddharth Bhat on February 26, 2020 at 07:30

The notation $Fun(C^{opp}, Sets)$ is never defined formally. Is this to be inferred through context?

Comment #5213 by Johan on June 11, 2020 at 09:38

@#4960: Well, it is defined later in Section 4.28. But we don't want to refer forward so I have removed the two instances where the notation was used before Section 4.28. Fixed here.

Comment #5811 by Zeyn Sahilliogullari on December 01, 2020 at 15:58

I think that "universal object" should be replaced by "universal element", since:

It seems this is the standard language, for example: https://ncatlab.org/nlab/show/universal+element

ξ is an element of a set, it is not clear to me which category it is an object of.

Comment #5812 by Johan on December 01, 2020 at 21:17

Often the way this comes up in algebraic geometry is that $\mathcal{C}$ is the category of schemes and $F$ sends a scheme $X$ to the set of isomorphism classes of certain kinds of geometric objects over $X$ and the terminology is chosen to make you think of this. For example $F$ might classify abelian schemes of relative dimension $g$ over $X$ with a polarization of degree $d$ and a full level $n$ structure. Then $F$ is representable, the object representing $F$ is the moduli scheme $A_{g, d, n}$ , and the universal object is the universal polarized abelian scheme with full level $n$ structure over $A_{g, d, n}$ .

Comment #6240 by Yuto Masamura on May 14, 2021 at 09:01

typo in the next paragraph of Exaple 4.3.4 (maybe too small one to point out): I think we should add a period after " $h:\mathcal C\to\mathit{PSh}(\mathcal C)$ ".

Comment #8427 by Elías Guisado on May 26, 2023 at 09:00

Maybe one could add to this section the definition of subfunctor (given in 26.15.3) stated in a general setting and also add the following remark: Let $\mathcal{C}$ be a category and let $f:A\to X$ be a morphism in $\mathcal{C}$ . Suppose $F:\mathcal{C}^\text{opp}\to\text{Sets}$ is a functor with a representation $\text{Mor}(-,X)\cong F$ . For an object $Y\in\text{Ob}(\mathcal{C})$ , define $F_f(Y)$ to be the image of the composite $\text{Mor}(Y,A)\xrightarrow{f_*}\text{Mor}(Y,X)\xrightarrow{\cong}F(Y)$ . Then $F_f$ is a subfunctor of $F$ and the natural transformation $\text{Mor}(-,A)\to F_f$ is an isomorphism if and only if $f$ is a monomorphism. Moreover, if these conditions hold and $\xi\in F(X)$ is the universal object for $\text{Mor}(-,X)\cong F$ , then $F(f)(\xi)\in F_f(A)$ is the universal object for $\text{Mor}(-,A)\cong F_f$ .

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4.3 Opposite Categories and the Yoneda Lemma

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