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X-MS-Exchange-CrossTenant-FromEntityHeader: Hosted X-MS-Exchange-CrossTenant-Id: 82c514c1-a717-4087-be06-d40d2070ad52 X-MS-Exchange-CrossTenant-MailboxType: HOSTED X-MS-Exchange-CrossTenant-UserPrincipalName: gH+7/Uiswrs/vo2aaM2RT9jT1OZL+qEtIUoQ+oeSL/bsVYsZxZda1HbgXET2vtVSPYwd8NZZs/430dWFBmRNpT/r/wkcvYjDb/BDkhTaIsY= X-MS-Exchange-Transport-CrossTenantHeadersStamped: SYBPR01MB5488 X-Mimecast-Spam-Score: 0 X-Mimecast-MFC-PROC-ID: Kb1dM9a_2rUsbgkjmxsCIzkA-q70gDvqhY1KTUzNejY_1737496303 X-Mimecast-Originator: mq.edu.au Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable 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=3D>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 writes: > Thanks, Steve, but I wrote [C^op,Set] to mean the category of all preshea= ves. =20 > > I didn=E2=80=99t know that this notation is sometimes used for the catego= ry of small presheaves (e.g. in Rosick=C3=BD=E2=80=99s paper). > > Best regards, > > Paul=20 > > =20 > > 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 cli= ck links or open attachments unless you recognise the sender and =20 > know the content is safe. =20 > > =20 > > Dear Paul,=20 > > =20 > > I agree that if H(-@a) is small then it does the job, but why should it b= e 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=E2=80=99s =E2=80=9CCartesian closed exact completi= ons=E2=80=9D. For various > generalizations, including the non-cartesian case, see Section 7 of my pa= per =E2=80=9CLimits of small functors=E2=80=9D with Brian Day (Example 7.4 = refers to the > Rosicky result).=20 > > =20 > > Of course if C is actually cartesian closed then C(-@a,b) is not just sma= ll but representable.=20 > > =20 > > As far as I can tell, in Saville=E2=80=99s thesis, the bicategory B corre= sponding to your C is itself supposed to be small (at least relative to Cat= ). > > =20 > > Best, > > =20 > > Steve. > > =20 > > On 21 Jan 2025, at 4:38=E2=80=AFAM, Paul Levy wrot= e: > > =20 > > Dear all, > > Let C be a locally small category. A functor C^op -> Set that=E2=80=99s= a colimit of representables is called a =E2=80=9Csmall presheaf=E2=80=9D. > > 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. Expli= citly, 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=E2=80=99s a 2-categorical version of (1) at the start of Sectio= n 6.2 of Saville=E2=80=99s thesis:=20 > > https://philipsaville.co.uk/thesis-for-screen.pdf=E2=9A=A0=EF=B8=8F > > Another related result is the cartesian closure of the category of conta= iners: > > https://pblevy.github.io/papers/hocont.pdf=E2=9A=A0=EF=B8=8F