Timeline for Yoneda embedding target
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2010 at 16:19 | comment | added | Mike Shulman | Harry, the point is that "Grothendieck topos" is the same as "category of sheaves on a site." Certainly once you know that, along with the meaning of "sheaf," then it follows trivially that any Grothendieck topos is a full subcategory of a presheaf category. But depending on your definition of "Grothendieck topos," the identification of such toposes with categories of sheaves might be definitional, or it might be Giraud's theorem. | |
| Jan 2, 2010 at 0:38 | comment | added | Harry Gindi | But Tom, that follows trivially from the fact that the category of sheaves on a site is a full subcategory of the category of presheaves. | |
| Jan 2, 2010 at 0:27 | comment | added | Tom Leinster | I don't think anyone quite answered Ilya's question. There are two slightly different things called toposes: elementary toposes (more general) and Grothendieck toposes (less general, but still including all presheaf categories). I'll use "topos" to mean "elementary topos". There is an accompanying notion of subtopos; a subtopos of a topos is a subcategory with certain properties. Now: it's a fact that a topos is Grothendieck iff it is a subtopos of some presheaf category. So if in your question "topos" means "Grothendieck topos", the answer is yes. | |
| Jan 1, 2010 at 20:57 | comment | added | Mike Shulman | Although, to be fair, Grothendieck was the original user of the word "topos," which was only later co-opted by Lawvere and Tierney for the more general notion of "elementary topos," necessitating the adjective "Grothendieck" to disambiguate the original definition. Many people quite justifiably still say "topos" to mean "Grothendieck topos". | |
| Jan 1, 2010 at 20:09 | comment | added | Harry Gindi | Mikael, rather. | |
| Jan 1, 2010 at 20:08 | comment | added | Harry Gindi | To clarify what Mike means, you're thinking of a "Grothendieck Topos", which is the sheaf category over any site (presheaf category is the sheaf category over the discrete site). In fact, all Grothendieck toposes are general toposes, but not conversely. | |
| Jan 1, 2010 at 20:04 | comment | added | Mikael Vejdemo-Johansson | Topoi are categories with a rich enough structure to capture set theory. It is a theorem that any presheaf category is a topos, though topoi do not have to be presheaves. At least not at first glance. I think there may well be a construction to make any (nice enough?) topos into a category of sheaves of some sort - but I am much too much of a beginner with topoi to tell you how. | |
| Jan 1, 2010 at 19:00 | comment | added | Ilya Nikokoshev | Now, here I start to have some memories. Is topos a certain subcategory of presheaf category? | |
| Jan 1, 2010 at 19:00 | vote | accept | Ilya Nikokoshev | ||
| Jan 1, 2010 at 18:55 | history | answered | Mike Shulman | CC BY-SA 2.5 |