Timeline for Category of representations of the path-groupoid
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11 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Sep 29, 2017 at 17:26 | comment | added | Carlos | In any case, my question remains. What replaces local systems in this non-flat case? | |
| Sep 29, 2017 at 17:25 | comment | added | Carlos | @DmitriPavlov Nice, thank you! In fact, it answers a "question" I had from the first comment, whether or not the category of representations of the path-groupoid encompasses more stuff than "just" vector bundles with connection. That is, any (smooth) functor $\mathcal{P}_1(X)\to \text{Vect}$ satisfies the appropriate descent data to assemble a vector bundle with connection (via parallel transport), and not just the transport functors from Schreiber & Waldorf. How are the results related? | |
| Sep 28, 2017 at 23:17 | comment | added | Dmitri Pavlov | Furthermore, in a forthcoming work with Stolz and Teichner I answer an n-dimensional version of your question: functors from the smooth path n-groupoid with thin homotopies as (n+1)-morphisms recover precisely bundles with n-connections. For n=1 one recovers the above results for vector and principal bundles with connection. For n>1 one recovers the n-category of bundle (n-1)-gerbes with connection. | |
| Sep 28, 2017 at 23:09 | comment | added | Dmitri Pavlov | In addition to the above-cited work of Schreiber and Waldorf, I would like to advertise arxiv.org/abs/1501.00967, which removes some technical assumptions of Schreiber and Waldorf (such as sitting instants) and proves a similar result. | |
| Sep 28, 2017 at 0:08 | comment | added | Carlos | However, I didn't mention this explicitly as I was trying to avoid influencing you going further into this direction. In the same way we have the web of correspondences {vector bundles with flat connection} ~ {representations of the fundamental groupoid} ~ {local systems}, we should(?) have a similar one when we lift flatness. The above is giving just one corner of this picture. What replaces local systems? It's not constructible sheaves as we know they give representations of the exit-path category (which can be further related to constructible vector bundles with flat connection). | |
| Sep 28, 2017 at 0:06 | comment | added | Carlos | As @DylanWilson guessed quite well, these functors encode the information of the parallel transport on bundles with (not necessarily flat) connection. And in a way, we could say that the category of representations of the path-groupoid is equivalent to the category of bundles with (not necessarily flat) connection. | |
| Sep 28, 2017 at 0:05 | comment | added | Carlos | @DavidRoberts I was thinking about representations in Vect. But, for starters, Sets would do. I am indeed aware of Schreiber & Waldorf's work (link) where they explore the so-called transport functors. I am under the impression that these only give a particular sector of the category of representations of $\mathcal{P}_1(X)$ satisfying further descent data, therefore missing "all the others". Correct me, please, if I'm wrong. | |
| Sep 27, 2017 at 21:33 | comment | added | David Roberts♦ | Which category are you asking for representations in? Vect? If so, then Dylan has it right. Similarly if reps are in GTors for G a Lie group (this is work of Schreiber and Waldorf) | |
| Sep 27, 2017 at 19:29 | comment | added | Dylan Wilson | Is the guess maybe "bundles equipped with a (not necessarily flat) connection"? But I don't think I understand what a 'thin homotopy' is well enough to stand by that claim. | |
| Sep 27, 2017 at 17:53 | history | asked | Carlos | CC BY-SA 3.0 |