Higher Order Yoneda?

Higher Order Yoneda?

[Christopher Townsend]

I was looking for a reference (or correction!) to the following observation
in indexed category theory.

Let E be a cartesian category and H an E-indexed category (that is H is a
functor from E^op to CAT, where CAT is some background category of possibly
large categories).

Then, if C is an internal category in E we have a categorical equivalence

Nat[Cat(_,C),H]=H(C_0)

where C_0 is the object of objects of C. The objects of Nat[Cat(_,C),H] are
the natural transformations and the morphisms are the modifications (see,
e.g. definition B1.2.1(c) in Johnstone's Elephant).

On objects, this equivalence is just Yoneda's lemma, so surely it has been
observed already that it extends to this 2-categorical statement?

Best wishes, Christopher Townsend

Re: Higher Order Yoneda?

[Ross Street]

[Note from Moderator: Apologies to Ross for the inadvertent delay in
posting this.]

>I was looking for a reference (or correction!) to the following observation
>in indexed category theory.

Try, for example, Theorem (5.15) of

13. Cosmoi of internal categories, Transactions American Math. Soc.
258 (1980) 271-318; MR82a:18007.

Regards,
Ross

PS Allow me to correct an annoyingly wrong gratuitous word on the
same page as that Theorem; the word "full" should be deleted on the
second line of (5.13).