Timeline for Topos associated to a category
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 15 at 7:32 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
deleted 9 characters in body
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| Apr 14 at 19:01 | history | edited | მამუკა ჯიბლაძე | CC BY-SA 4.0 |
some very strange mathjax thing
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| Jan 11, 2012 at 9:10 | vote | accept | Marc Nieper-Wißkirchen | ||
| Jan 11, 2012 at 0:49 | answer | added | Steve Lack | timeline score: 27 | |
| Jan 10, 2012 at 21:44 | comment | added | Marc Nieper-Wißkirchen | @Martin: For the moment, the analogy is just formal to me. For the topos that I write $\mathbb{A}$ Mac Lane and Moerdijk write $S[U]$ in analogy to a polynomial ring because the set of morphisms from the polynomial algebra over a ground ring to another algebra is just the set of elements of that other algebra as the category of morphisms from a topos to $S[U]$ is the category of objects of that topos. I doesn't like that notation too much as one has to turn arrows around so that one should introduce, at least formally, somewhere a $\operatorname{Spec}$. | |
| Jan 10, 2012 at 21:32 | comment | added | Marc Nieper-Wißkirchen | @Sergio: If you have a topos, which is a category of a certain form, and you forget about that extra information, you are left with just a category, which I call the category of objects of the topos. It happens to be a locally presentable category [Francis Borceux: Handbook of Categorical Algebra: Categories of Sheaves (proposition 3.4.16)]. | |
| Jan 10, 2012 at 17:38 | comment | added | Buschi Sergio | PLease, let me know what is the "locally presentable category of objects in a topos E" (?) | |
| Jan 10, 2012 at 17:34 | comment | added | Martin Brandenburg | I don't really get the analogy with algebraic geometry, except that we use the same symbols ($\mathbb{A}$, $\mathcal{O}$, $f^*$ ...). How strict is this analogy really? | |
| Jan 10, 2012 at 12:04 | comment | added | Zoran Skoda | There is some hesitation between terminology continuous and cocontinuous. Bass used right and left continuous and right continuous was more important in module theory, so some retained continuous for what pure category theorists say cocontinuous. See also Lurie's book which also has it that way (and also Rosenberg who follows Bass). | |
| Jan 10, 2012 at 11:55 | history | edited | Marc Nieper-Wißkirchen | CC BY-SA 3.0 |
added the attributes "locally presentable" and "cocontinuous"
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| Jan 10, 2012 at 11:53 | comment | added | Marc Nieper-Wißkirchen | @Tom: You are right; in my question I am a bit sloppy when it comes to size issues. @Andrej: Do you possibly mean cocontinuous? I will changed my question to address both comments in a manner that is hopefully helpful. | |
| Jan 10, 2012 at 11:18 | comment | added | David Roberts♦ | I like your question. But I'll let others more expert than me answer. | |
| Jan 10, 2012 at 10:32 | comment | added | Andrej Bauer | You should be willing to change one of the $\mathrm{Hom}$ to something else, such as "continuous functors". But I'll let others more expert than me answer. | |
| Jan 10, 2012 at 9:35 | comment | added | Tom Leinster | I guess you want to cut down to small categories C, in order to get a Spec(C) satisfying your requirements. When C is small, the theory of diagrams on C is a geometric theory, and therefore has a classifying topos Spec(C). So (modulo the size question), I think the answer to your question must be yes. But I'll let others more expert than me answer. | |
| Jan 10, 2012 at 9:03 | history | asked | Marc Nieper-Wißkirchen | CC BY-SA 3.0 |