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 Date: Tuesday, 21 January 2025 at 00:11 To: Paul Levy (Computer Science) Cc: Categories mailing list 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 know the content is safe. 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 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 You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups