Timeline for References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2023 at 15:19 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: The only source I am aware of is Section 5.3.2 in Lurie's Higher Topos Theory. | |
| Mar 26, 2023 at 15:07 | comment | added | მამუკა ჯიბლაძე | Great, thanks! Where would you recommend to read about ∞-flatness? | |
| Mar 26, 2023 at 15:03 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: Yes, I see now. The notion described in my answer is essentially the homotopy coherent analogue of the construction described in Johnstone's book. This is also the construction used by Moerdijk in his book “Classifying Spaces and Classifying Topoi”, and also by Weiss in his paper. Roughly, a $\def\bC{{\bf C}}\bC$-bundle over $X$ is an ∞-flat functor $\def\Sh{{\sf Sh}}\bC→\Sh(X)$, where $\Sh(X)$ is the ∞-topos of ∞-sheaves on $X$. To extract such a functor from the above construction, send $A∈\bC$ to the ∞-sheaf given by mapping the Čech cocycle data into $A$. | |
| Mar 26, 2023 at 13:15 | comment | added | მამუკა ჯიბლაძე | Yes it is quite old. Here $\mathcal E$ is a (Grothendieck) topos. It is the content of Diaconescu's theorem that the category of geometric morphisms from $\mathcal E$ to the presheaf topos of $\mathbf C$ is equivalent to the category of such torsors. It may be found already in Johnstone's first (1977) topos theory book, Theorem 4.34 there (page 113). | |
| Mar 26, 2023 at 13:11 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: Inversion of morphisms only happens once you pass to the space of $\bf C$-bundles. If you consider a single $\bf C$-bundle, then it certainly sees noninvertible morphisms, as indicated above. What is $\cal E$ in the above comment? Is there a reference for this notion of a bundle? | |
| Mar 26, 2023 at 13:07 | comment | added | მამუკა ჯიბლაძე | I see, thanks. One more question. I am aware of another notion of principal bundle that does not force to invert anything, do you know if it is related? I mean the version in "ordinary" category theory: a $\mathbf C$-torsor in $\mathcal E$ is a functor $\mathbf C\to\mathcal E$ whose Grothendieck construction is a (co?)filtered internal category of $\mathcal E$. | |
| Mar 26, 2023 at 13:00 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: For example, take an open cover with two elements $\{U,V\}$. The the modified Čech nerve is the category internal to topological spaces whose nonidentity morphisms look like $U←U∩V→V$, and are not invertible. | |
| Mar 26, 2023 at 12:51 | comment | added | მამუკა ჯიბლაძე | Oh wow that's interesting! Thanks. This seems to be a new notion, have not seen it before. So this modified nerve is not equivalent to any kind of groupoid? | |
| Mar 26, 2023 at 12:45 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: No, it is not. It is a functor from the modified Čech nerve of $\{U_i\}_{i∈I}$ to $\bf C$. The latter is introduced in Definition 6.1 and has the disjoint union of all finite intersections as objects, and their inclusions as morphisms. | |
| Mar 26, 2023 at 12:42 | comment | added | მამუკა ჯიბლაძე | Is not what you describe a functor from the nerve of $\{U\}$ to $\mathbf C$? | |
| Mar 26, 2023 at 12:39 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: No, it's the other adjoint (invert morphisms up to homotopy, not discard the noninvertible ones). This is natural since this is what the functor $B$ does anyway. | |
| Mar 26, 2023 at 12:36 | comment | added | მამუკა ჯიბლაძე | So a principal $\mathbf C$-bundle is the same thing as a principal $\operatorname{iso}(\mathbf C)$-bundle, where $\operatorname{iso}(\mathbf C)\to\mathbf C$ is universal among ∞-groupoids mapping to $\mathbf C$? | |
| Mar 26, 2023 at 12:26 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |