category theory

Yoneda lemma

# Contents

## Idea

For $C$ a locally small category, every object $X$ of $C$ induces a presheaf on $C$: the representable presheaf $h_X$ represented by $X$. This assignment extends to a functor $C \to [C^{op}, Set]$ from $C$ to its category of presheaves. The Yoneda lemma implies that this functor is full and faithful and hence realizes $C$ as a full subcategory inside its category of presheaves.

Recall from the discussion at representable presheaf that the presheaf represented by an object $X$ of $C$ is the functor $h_X :C^{op} \to Set$ whose assignment is illustrated by

which sends each object $U$ to $Hom_C(U,X)$ and each morphism $\alpha:U'\to U$ to the function

$h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).$

Moreover, for $f : X \to Y$ an morphism in $C$, this induces a natural transformation $h_f : h_X \to h_Y$, whose component on $U$ in $X$ is illustrated by

For this to be a natural transformation, we need to have the commuting diagram

$\array{ h_X U & \stackrel{h_f U}{\rightarrow} & h_Y U \\ \mathllap{h_X\alpha\quad}{\downarrow} & {} & \mathrlap{\downarrow}{\quad h_Y\alpha} \\ h_X U' & \stackrel{h_f U'}{\rightarrow} & h_Y U' }$

but this simply means that it doesn’t matter if we first “comb” the strands back to $U'$ and then comb the strands forward to $Y$, or comb the strands forward to $Y$ first and then comb the strands back to $U'$

which follows from associativity of composition of morphisms in $C$.

## Definition

The Yoneda embedding for $C$ a locally small category is the functor

$Y : C \to [C^{op}, Set]$

from $C$ to the category of presheaves over $C$ which is the image of the hom-functor

$Hom : C^{op} \times C \to Set$

under the Hom adjunction

$Hom(C^{op} \times C , Set) \simeq Hom(C, [C^{op}, Set])$

Hence $Y$ sends any object $c \in C$ to the representable presheaf which assigns to any other object $d$ of $C$ the hom-set of morphisms from $d$ into $c$:

$Y(c) : C^{op} \stackrel{C(-,c)}{\to} Set \,.$

## Properties

It follows from the Yoneda lemma that the functor $Y$ is full and faithful. It is also limit preserving (= continuous functor), but does in general not preserve colimits.

The Yoneda embedding of a small category $S$ into the category of presheaves on $S$ gives a free cocompletion of $S$.

If the Yoneda embedding of a category has a left adjoint, then that category is called a total category .

Revised on November 11, 2014 07:19:25 by Tim Porter (2.31.52.236)