exponentiating by a small presheaf

[Paul Levy <p.b.levy@bham.ac.uk>]

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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/WxUqCNLJxki0EqxqRSmfVSyu55n?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/w28RCOMK7Ycp0LXLGFvhDSG3y5O?domain=pblevy.github.io>



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