Timeline for Are sieves in locally small categories still sets?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Sep 9, 2010 at 19:37 | comment | added | Aleks Kissinger | Great! That clears it up. | |
| Sep 8, 2010 at 4:19 | comment | added | Mike Shulman | Perhaps the objection was that M&M specifically say (on p37) that a sieve consists of a downward-closed set of arrows. However, note that on p36 the category C was introduced as "an arbitrary small category", so in the context they were working in, a sieve actually is a set. Furthermore, according to their "preliminaries" section, M&M are using the "one universe" foundation according to which even large categories are "sets" (just "large sets") rather than proper classes. | |
| Sep 7, 2010 at 23:13 | comment | added | Todd Trimble | In most applications, where one speaks of sieves and covering sieves (with respect to a topology), the underlying category $C$ of a site is assumed to be small. Otherwise, the sieve $S$ won't actually be a set, but a class (as you observe). Which isn't necessarily problematic; it depends what you want to do. | |
| Sep 7, 2010 at 21:03 | comment | added | Eric Finster | I think your observation about a category with a terminal object is correct. You could even that $C = \mathcal Set$ itself as an example. Do you have some objection to a sieve being a class of arrows closed under precomposition? | |
| Sep 7, 2010 at 20:38 | history | asked | Aleks Kissinger | CC BY-SA 2.5 |