nLab
Yoneda embedding

Contents

Idea

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

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

which sends each object UU to Hom C(U,X)Hom_C(U,X) and each morphism α:UU\alpha:U'\to U to the function

h Xα:Hom C(U,X)Hom C(U,X).h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).

Moreover, for f:XYf : X \to Y an morphism in CC, this induces a natural transformation h f:h Xh Yh_f : h_X \to h_Y, whose component on UU in XX is illustrated by

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

h XU h fU h YU h Xα h Yα h XU h fU h YU\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 UU' and then comb the strands forward to YY, or comb the strands forward to YY first and then comb the strands back to UU'

which follows from associativity of composition of morphisms in CC.

Definition

The Yoneda embedding for CC a locally small category is the functor

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

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

Hom:C op×CSet Hom : C^{op} \times C \to Set

under the Hom adjunction

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

in the closed symmetric monoidal category Cat.

Hence YY sends any object cCc \in C to the representable presheaf which assigns to any other object dd of CC the hom-set of morphisms from dd into cc:

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

Properties

It follows from the Yoneda lemma that the functor YY 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 SS into the category of presheaves on SS gives a free cocompletion of SS.

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

Revised on August 6, 2014 12:07:29 by Colin Zwanziger (174.63.87.107)