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Cleaning or creating entries related to corings (e.g. grouplike element, Sweedler coring) and entwining structures, including personal entries Gabriella Böhm, Tomasz Brzeziński etc. On the edge of this activity I am interested in the relation between classical correspondence between flat connections and the descent data in abelian context; it could be related to the theorem of Urs and Konrad on the relation between descent data and transport functors in global context. I would like to know the parallel precisely.
it could be related to the theorem of Urs and Konrad on the relation between descent data and transport functors in global context. I would like to know the parallel precisely.
I still don’t quite know what I should check against.
What I have is a general remark: both flat structures = local systems as well as cocycles and descent data may be thought of as nothing but morphisms out of certain $\infty$-groupoids. As such they look every much the same and are very much the same.
I do not talk about local systems, but about the descent data in abelian context. Flat structures = local systems may be about integration. Local systems are global and flat Koszul connection is a local structure. You can both understand locally. But the descent data are something about the relation of two fibers in a fibered category. Now in your case you have a fibered category of bundles with flat connection. Fibered category descent under Benabou-Raubaud can be related to monadic or comonadic descent. Corings give a special case of comonadic descent. For comonads with grouplike (e.g. for Sweedler coring) one has Amitsur complex and a Koszul connection.
I do not see that in your case one also has a grouplike for the corresponding comonad and if it is good to look via comonad or it is better to look via fibered category.
out of certain ∞-groupoids
This is all nice another approach. I am in dimension one. I do not care about infinite dimensional generalization, just the classical case. The general nonsense on corings with group like elements (and by Menini somewhat more general) gives sense to Koszul connections. Now, I want this to be the special case of the same theorem. So we are in the picture of Grothendieck’s crystals.
You see, you look at certain groupoids. This is a special feature in the global theory of connections. I want to say that the usual connections (as differential forms) are local objects and the picture depends only on the de Rham descent, treated as any other comonadic descent. Just easy genarl nonsense gives relation between descent data and flat Koszul connections. It is pure algebra. Now sometimes in some context one can do integration. For example in some differential context. Then one can globalize. But I want to globalize the formal result sticking to formal description I have at the formal level of comonads.
Well, the de Rham descent I can also understand as being homs out of an $\infty$-Lie groupoid, one that happens to have infinitesimal morphisms.
But okay. I understand you are after an algebraic expression. But help me: what exactly is the algebraic expression of which you would like to know its relation to flat connections?
The Roiter’s theorem says: the semifree dgas are in correspondence with corings with a group-like element. Moreover (addition, due Brzezinski etc.) flat connections for a semi-free dga are in 1-1 correspondence with the comodules over the corresponding coring with a group-like element (see semifree dga).
Comodules over a coring – this is a clean description of descent data in additive comonadic setup. Such comonadic descent can be viewed along some morphism in a (bi)fibered category of rather general kind.
Now you have a fibered category. In fact you have a stack of bundles with flat connection. How can you relate the connection you work with to the flat connection on semifree dga in such a way that the corresponding comonadic descent along given cover is exactly the descent you produce with Konrad ? I want your theorem with Konrad to be either a special case of the addition to the Roiter’s theorem or possibly of a parallel e.g. integrated statement.
I don’t think the stuff I did with Konrad and John is of particular relevance here. The point of the theorem you mention is to identify smooth transport over finite dimensional paths with connection data.
But here for connections over corings, everything is purely algebraic and infinitesimal.
I do understand the notion of connection on a dga that is described at connection for a coring. That’s the notion from Lie infinity-algebroid representations: because if we think of these dgas as being Chevalley-Eilenberg dgs of $\infty$-Lie algebroids, then these “connections” or “superconnections” are representations of these. In the case of the tangent Lie algebroid (dually: de Rham dga) this are flat connections on complexes of vector bundles. Under mild conditions this is the same as de Rham descent and D-modules etc.
What I still don’t have a good feeling for (because I haven’t thought about it) is what the dga associated to a coring means.
I did not mention John. I was talking about your article with Konrad only with the statement that the descent data coprrespond to certain transport functors.
because if we think of these dgas as being Chevalley-Eilenberg dgs of ∞-Lie algebroids, then these “connections” or “superconnections” are representations of these
You see you keep going into elaborated situations with lots of special circumtances, and I want to simplify to simple common denominator. WHATEVER situation with the 1-categorical descent is, it is described via fibered categories. For a single cover usually it boils down to monadic or comonadic descent. If we work with vector bundles then we can also add and subtract, so we deal with additive comonads. In infiniitesimal case, even for principal bundles, the linearization gives additive comonads. So whatever we start with we want to look from the point of view of (co)monads.
Now in infintesimal case we have grouplike element and we have additivity, so we get the situation with Roiter’s theorem and with appendix for comodules and flat connections. This is true in arbitrary characteristic. Now how to integrate this statement in characteristic zero to get integrated statement about the (co)monadic descent which is equivalent to the descent data in your paper with Konrad ? More generally when the appendix to Roiter’s theorem can be integrated ? I suspect a clean general setup for this exists.
That’s the notion from Lie infinity-algebroid representations: because if we think of these dgas as being Chevalley-Eilenberg dgs of ∞-Lie algebroids, then these “connections” or “superconnections” are representations of these.
There is non finite-dimensionality assumption, nor the ground ring needs to be a field for Amitsuir complex to make sense. Corings are taken over an arbitrary ring, where nothing like Koszul duality can be used. It is nice of course that it fits with the connections for Lie algebroids, but this is not the abstract nonsense situation. The abstract nonsense is at the level of comonads.
Under mild conditions this is the same as de Rham descent and D-modules etc.
The case of principal bundles with connection is already much more special than working with comonadic additive descent with grouplike condition; to have more conditions means going further from basic general nonsense. The case of corings, a bit more special than for comonads, is still quite general – for example the connections in the sense of Connes and Karoubi in noncommutative geometry fit in. While there is no satisfactory notion of Lie infinity groupoid in noncommutative context!
I am not sure what else to say. The whole principal nonabelian higher Cech descent that I have been considering does not admit a comonadic description as it deals with just fibered oo-groupoids, not bifibered ones.
This is an artifact of the particular formulation of the Benabou-Raubaud theorem. The way you have discussed it, e.g. at Sweedler coring shows that you do consider also pushforwards for particular morphisms. It is not important to have bifibered category, it is in a way sufficient to have adjoint for the particular inverse image functor corresponding to the cover. It is difficult to give examples of descent which are not (co)monadic. On th eother hand, I am interested in algebaric categories in which the analogue of the fundamental group are defined using Tannakian reconstruction and where we do not have genuine paths and where locally we still have differential forms and thickenings. Infinity analogues are important in the new brave algebra. There is no question that infinitesimal analogue both in smooth and algebraic category have the correspondence between flat connectiosn and descent data. Now globalizations are of various kinds, and to understand systematically we should directly try to integrate this pictrue in abstract context.
Besides I was thinking about transport for associated vector bundles.
I expanded references at connection for a coring.
New entry entwined module. The reference and comment on van Osdol 1973 moved from Hopf module (which I expanded otherwise) to entwined module.
I updated slightly this stub in wake of new edits at the page distributive law concerning the closely related mixed distributive laws. It needs much more, but it will wait as I need to finish a review of the paper by Porta and Sala on 2-dimensional categorical Hall algebra for Zbl.
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