enriched Yoneda lemma



The enriched Yoneda lemma is the generalization of the usual Yoneda lemma from category theory to enriched category theory.


We discuss here two forms of the Yoneda lemma.

Let VV be a (locally small) closed symmetric monoidal category, so that VV is enriched in itself via its internal hom.

Weak form

A weak form of the enriched Yoneda lemma says that given a VV-enriched functor F:CVF: C \to V and an object cc of CC, the set of VV-enriched natural transformations α:hom C(c,)F\alpha: \hom_C(c, -) \to F is in natural bijection with the set of elements of F(c)F(c), i.e., the set of morphisms IF(c)I \to F(c), obtained by composition:

I1 chom C(c,c)αcF(c)I \stackrel{1_c}{\to} \hom_C(c, c) \stackrel{\alpha c}{\to} F(c)

Strong form

Now suppose that VV is in addition (small-)complete (has all small limits). Then, given a small VV-enriched category CC and VV-enriched functors F,G:CVF, G: C \to V, one may construct the object of VV-natural transformations as an enriched end:

V C(F,G)= cV(F(c),G(c))V^C(F, G) = \int_c V(F(c), G(c))

(which may in turn be expressed as an ordinary limit in VV). This is the hom-object in the enriched functor category.

A strong form of the enriched Yoneda lemma specifies a VV-natural isomorphism

V C(hom C(c,),F)F(c).V^C(\hom_C(c, -), F) \cong F(c).

This implies the weak form by applying the functor hom(I,):VSet\hom(I, -): V \to Set.


The weak form is in section 1.9, the strong form in section 2.4 of

  • Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982) (pdf)

Generalization to the case that the enriching monoidal category is not closed or symmetric is in

  • Vladimir Hinich, Enriched Yoneda lemma, Theory and Applications of Categories, Vol. 31, 2016, No. 29, pp 833-838 (TAC)

Last revised on September 1, 2016 at 14:31:20. See the history of this page for a list of all contributions to it.