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82c514c1-a717-4087-be06-d40d2070ad52:0 X-MS-Exchange-SkipListedInternetSender: ip=[52.101.196.90];domain=LO3P265CU004.outbound.protection.outlook.com X-MS-Exchange-ExternalOriginalInternetSender: ip=[52.101.196.90];domain=LO3P265CU004.outbound.protection.outlook.com X-MS-Exchange-Transport-CrossTenantHeadersStripped: ML1PEPF0000F177.ausprd01.prod.outlook.com X-MS-PublicTrafficType: Email X-MS-Office365-Filtering-Correlation-Id-Prvs: 3887f106-7bb1-4d2a-5600-08dd3a059762 X-Moderation-Data: 1/21/2025 10:25:39 AM X-LD-Processed: 82c514c1-a717-4087-be06-d40d2070ad52,ExtAddr,ExtAddr X-Auto-Response-Suppress: DR, OOF, AutoReply X-MS-Exchange-CrossTenant-Network-Message-Id: 498fa3dd-d881-4961-6dd0-08dd3a059c76 X-MS-Exchange-CrossTenant-Id: 82c514c1-a717-4087-be06-d40d2070ad52 X-MS-Exchange-CrossTenant-AuthSource: ML1PEPF0000F177.ausprd01.prod.outlook.com X-MS-Exchange-CrossTenant-AuthAs: Anonymous X-MS-Exchange-CrossTenant-FromEntityHeader: Internet X-MS-Exchange-CrossTenant-OriginalArrivalTime: 21 Jan 2025 10:25:40.9525 (UTC) X-MS-Exchange-Transport-CrossTenantHeadersStamped: SY8PR01MB8997 X-MS-Exchange-UnifiedGroup-DisplayName: Categories mailing list X-MS-Exchange-UnifiedGroup-Address: categories@mq.edu.au X-MS-Exchange-UnifiedGroup-MailboxGuid: 9c2f954e-92a7-451b-b723-a07075d7adb5 X-MS-Exchange-UnifiedGroup-CustomizedMessage: RedeemedBusinessGuests X-MS-Exchange-Parent-Message-Id: Auto-Submitted: auto-generated X-MS-Exchange-Generated-Message-Source: Throttled Fork Delivery Agent X-OriginatorOrg: mq.edu.au --_000_CWLP265MB4190FBE062297AFB6EEAC06BE6E62CWLP265MB4190GBRP_ Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Dear Paul, It isn't exactly the result you quote, but there's an old paper by H. Enge= nes (Subobject classifiers and classes of subfunctors, Math. Scand. 34 (197= 4), 145=97152) where he observes that if C is a category such that each sli= ce C/B is equivalent to a small category (even though C itself may not be s= mall) than [C^op,Set] is a topos. Best regards Peter ________________________________ From: Paul Levy Sent: Monday, January 20, 2025 5:38 PM To: 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=92s a colimi= t of representables is called a =93small presheaf=94. 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. Explicit= ly, 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 cas= e of a representable presheaf? Best regards, Paul PS there=92s a 2-categorical version of (1) at the start of Section 6.2 of = Saville=92s thesis: https://philipsaville.co.uk/thesis-for-screen.pdf Another related result is the cartesian closure of the category of containe= rs: https://pblevy.github.io/papers/hocont.pdf You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --_000_CWLP265MB4190FBE062297AFB6EEAC06BE6E62CWLP265MB4190GBRP_ Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable
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=97152) where he observes that if C is a category such that eac= h 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.lev= y@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=92s a colimit = of representables is called a =93small presheaf=94.

Here are two observations.

  1. More generally, let C be monoida= l.  In the multicategory [C^op, Set], any small presheaf is exponentia= ting.

To see (1), it suffices to prove it for a representable preshe= af.  Explicitly, a presheaf H exponentiated by C(-,a) is H(- * a).&nbs= p; The construction of (2) is similar.

Has either result appeared in the literature?  At least f= or the special case of a representable presheaf?

Best regards,

Paul

PS there=92s a 2-categorical version of (1) at the start of Se= ction 6.2 of Saville=92s thesis:&nb= sp;

https://philip= saville.co.uk/thesis-for-screen.pdf

Another related result is the cartesian closure of the categor= y of containers:

https://pblevy.github.io= /papers/hocont.pdf

 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View group files   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
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