Re: exponentiating by a small presheaf

["P.T. Johnstone" <ptj1000@cam.ac.uk>]

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Dear Paul,

It isn't exactly the result you quote,  but there's an old paper by H. Engenes (Subobject classifiers and classes of subfunctors, Math. Scand. 34 (1974), 145—152) where he observes that if C is a category such that each slice C/B is equivalent to a small category (even though C itself may not be small) than [C^op,Set] is a topos.

Best regards
Peter
________________________________
From: Paul Levy <p.b.levy@bham.ac.uk>
Sent: Monday, January 20, 2025 5:38 PM
To: categories@mq.edu.au <categories@mq.edu.au>
Subject: exponentiating by a small presheaf


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://url.au.m.mimecastprotect.com/s/izjQCMwGj8CqRMyxGtwfzS89wi1?domain=philipsaville.co.uk>

Another related result is the cartesian closure of the category of containers:

https://pblevy.github.io/papers/hocont.pdf<https://url.au.m.mimecastprotect.com/s/a-acCNLJxki0EDqV1t4hVSy4SJT?domain=pblevy.github.io>



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