enriched Yoneda lemma
In higher category theory
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 be a (locally small) closed symmetric monoidal category, so that is enriched in itself via its internal hom.
A weak form of the enriched Yoneda lemma says that given a -enriched functor and an object of , the set of -enriched natural transformations is in natural bijection with the set of elements of , i.e., the set of morphisms , obtained by composition:
Now suppose that is in addition (small-)complete (has all small limits). Then, given a small -enriched category and -enriched functors , one may construct the object of -natural transformations as an enriched end:
(which may in turn be expressed as an ordinary limit in ). This is the hom-object in the enriched functor category.
A strong form of the enriched Yoneda lemma specifies a -natural isomorphism
This implies the weak form by applying the functor .
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)
Revised on September 1, 2016 14:31:20
by Urs Schreiber