Re: exponentiating by a small presheaf

[Richard Garner <richard.garner@mq.edu.au>]

Hi Paul,

This is kind-of related. Defining the category of small presheaves
typically requires first defining the non-locally-small category of all
presheaves, and then taking within it the closure of the representables
under small colimits.

However you can avoid this. Define any arbitrary presheaf X to be
moderate if, for all presheaves Y, the set of natural transformations
X=>Y is small. The category of moderate presheaves is locally small.
Moreover, it clearly contains the representables, and has small colimits
computed pointwise. So you can define the category of small presheaves
as the closure of the representables under small colimits in the locally
small category of moderate presheaves. (It would seem reasonable to
think that this is the whole category, but it seems easiest to avoid
having to find out if this is true.)

The same argument works for V-categories, avoiding the need for
universe-enlarging V to V' in order to define the category of small
presheaves.

All the best,

Richard



Paul Levy <p.b.levy@bham.ac.uk> writes:

> Thanks, Steve, but I wrote [C^op,Set] to mean the category of all presheaves.  
>
> I didn’t know that this notation is sometimes used for the category of small presheaves (e.g. in Rosický’s paper).
>
> Best regards,
>
> Paul 
>
>  
>
> From: Steve Lack <steve.lack@mq.edu.au>
> Date: Tuesday, 21 January 2025 at 00:11
> To: Paul Levy (Computer Science) <p.b.levy@bham.ac.uk>
> Cc: Categories mailing list <categories@mq.edu.au>
> Subject: Re: exponentiating by a small presheaf
>
>  CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and  
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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⚠️
>
>  Another related result is the cartesian closure of the category of containers:
>
>  https://pblevy.github.io/papers/hocont.pdf⚠️