Timeline for Topos-theoretic Galois theory
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 12, 2022 at 13:13 | comment | added | user1022117 | @მამუკაჯიბლაძე Thanks for your comments! :-) | |
| Feb 11, 2022 at 14:51 | comment | added | მამუკა ჯიბლაძე | Since in particular $G_0\to X$ is a local homeomorphism, there is a forgetful functor $U:S(X)\to X$. Dito considers the case when $X$ admits the notion of a discrete category - by definition this means that $U$ has a left adjoint right inverse $F$. Then $F(1)$ has all good properties of the universal cover for $X$, and in good cases agrees with it. In particular, for a point $x:1\to X$ one obtains a toposophic groupoid $x^*F(1)$ which again in nice enough cases turns out to be a discrete groupoid with unique object and automorphism group equal to the fundamental group of $X$. | |
| Feb 11, 2022 at 14:44 | comment | added | მამუკა ჯიბლაძე | Still another approach by the late Dito Pataraia mighr also interest you. In "Internal categories in a left exact cosimplicial category" he considers, for a topos $X$, toposophic groupoids $G$ together with a morphism of simplicial toposes $N(G)\to ad(X)$ where $N(G)$ is the nerve of $G$ and $ad(X)$ is the antidiscrete simplicial topos (with the topos of $n$-simplices equal to $X^{n+1}$), such that constituting geometric morphisms $N(G)_n\to X^{n+1}$ are local homeomorphisms. These, with obvious morphisms, form a category $S(X)$. | |
| Feb 11, 2022 at 14:27 | comment | added | მამუკა ჯიბლაძე | You might also find useful "What is the fundamental group?" by J. F. Kennison. In that paper, for a connected locally connected locale $X$, to each connected base $\mathcal V$ of $X$ a sheaf of groups $\pi_{\mathscr V}$ on $X$ is assigned such that, given a point $x:1\to X$, the inverse limit over all $\mathscr V$ of groups $x^*\pi_{\mathscr V}$ gives the fundamental progroup of $X$. In nice cases it agrees with the "usual" fundamental group. | |
| Feb 11, 2022 at 12:17 | comment | added | user1022117 | @მამუკაჯიბლაძე Very nice, thanks! :-) | |
| Feb 11, 2022 at 10:50 | comment | added | მამუკა ჯიბლაძე | He has prodiscrete version, see Theorem 3.4.1 (page 26) | |
| Feb 8, 2022 at 17:23 | comment | added | user1022117 | Does Dubuc prove $(\ast)$? | |
| Feb 8, 2022 at 16:57 | comment | added | მამუკა ჯიბლაძე | The (pro)finite version, to be precise. And I now recalled, more general version can be also found in "On the representation theory of Galois and Atomic Topoi" by Dubuc | |
| Feb 8, 2022 at 16:06 | comment | added | user1022117 | @მამუკაჯიბლაძე Is that the fundamental group that is used in $(\ast)$? Moerdijk's comment suggests there are several variants of "fundamental group of a topos". | |
| Feb 8, 2022 at 16:06 | comment | added | user1022117 | @Andry Thanks for the references! | |
| Feb 7, 2022 at 11:27 | comment | added | Benjamin Steinberg | I first learned of fundamental groups of topoi from numdam.org/item/?id=CTGDC_1981__22_3_301_0 | |
| Feb 7, 2022 at 5:59 | history | became hot network question | |||
| Feb 7, 2022 at 5:14 | answer | added | Alec Rhea | timeline score: 7 | |
| Feb 7, 2022 at 1:40 | answer | added | user234212323 | timeline score: 6 | |
| Feb 6, 2022 at 19:37 | comment | added | user1022117 | Ah, using Google translator I think Exercice 2.7.5 is exactly the statement $(\ast)$. | |
| Feb 6, 2022 at 19:11 | comment | added | Andry | Let me start by a disclaimer, I am no expert in the subject discussed nor am I working in the field, just a curious fellow who was intrigued by those beautiful concepts once and remember a couple of references. Borceux and Janelidze's book "Galois Theories", might be a good place to look. I also remember Olivia Caramello (perhaps with Laurent Lafforgue) did some work in the subject, a simple google search should give you something. There is also an old paper of Barr and Diaconescu "On locally simply connected toposes and their fundamental groups". | |
| Feb 6, 2022 at 18:54 | comment | added | Piotr Achinger | Grothendieck's Galois theory is limited to finite covering spaces i.e. locally constant sheaves of finite sets. I don't know for which topoi the category of locally constant sheaves of finite sets is a Galois category in Grothendieck's sense. More generally, there is a notion of a (tame) infinite Galois category due to Bhatt and Scholze. I think they show an example of a topos such that locally constant sheaves do not form a tame infinite Galois category. I am sure you can always define the "shape" of a topos as a pro-homotopy type, but I don't know how its $\pi_1$ relates to local systems. | |
| Feb 6, 2022 at 18:48 | comment | added | მამუკა ჯიბლაძე | Have you checked 8.4 of Johnstone's (first, 1977) topos theory book? | |
| Feb 6, 2022 at 18:11 | history | asked | user1022117 | CC BY-SA 4.0 |