Re: exponentiating by a small presheaf

[Steve Lack <steve.lack@mq.edu.au>]

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Dear Paul,

I agree that if H(-@a) is small then it does the job, but why should it be small? Take H to be the representable C(-,b); then this would say that C(-@a,b) is small. If it is small, then the general case follows. This is Proposition 1 of Rosicky’s “Cartesian closed exact completions”. For various generalizations, including the non-cartesian case, see Section 7 of my paper “Limits of small functors” with Brian Day (Example 7.4 refers to the Rosicky result).

Of course if C is actually cartesian closed then C(-@a,b) is not just small but representable.

As far as I can tell, in Saville’s thesis, the bicategory B corresponding to your C is itself supposed to be small (at least relative to Cat).

Best,

Steve.


On 21 Jan 2025, at 4:38 AM, Paul Levy <p.b.levy@bham.ac.uk> wrote:

Dear all,
Let C be a locally small category.  A functor C^op -> Set that’s a colimit of representables is called a “small presheaf”.
Here are two observations.

  1.  Let C be cartesian.  In the cartesian category [C^op, Set], any small presheaf is exponentiating.
  2.  More generally, let C be monoidal.  In the multicategory [C^op, Set], any small presheaf is exponentiating.

To see (1), it suffices to prove it for a representable presheaf.  Explicitly, a presheaf H exponentiated by C(-,a) is H(- * a).  The construction of (2) is similar.
Has either result appeared in the literature?  At least for the special case of a representable presheaf?
Best regards,
Paul
PS there’s a 2-categorical version of (1) at the start of Section 6.2 of Saville’s thesis:
https://philipsaville.co.uk/thesis-for-screen.pdf<https://philipsaville.co.uk/thesis-for-screen.pdf>
Another related result is the cartesian closure of the category of containers:
https://pblevy.github.io/papers/hocont.pdf<https://pblevy.github.io/papers/hocont.pdf>


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