Slides: On Nonabelian Differential Cohomology
Posted by Urs Schreiber
On nonabelian differential cohomology
(52 pdf slides)
This is supposed to be one way to motivate the definition of -connections (pdf, blog, arXiv) along the lines discussed in (Generalized) Differential Cohomology and Lie Infinity-Connections.
Some topics are only mentioned rather briefly in these slides:
For more on the smooth spaces and smooth classifying spaces for -valued forms, see Space and Quantity.
For more on the nature of see The inner automorphism 3-group of a strict 2-group.
For more on the functorial description of connections used here, see The first edge of the cube.
For more on the relation between smooth 2-functors and Lie 2-algebra valued forms, see Smooth 2-functors and differential forms.
Question on simplicial -categories.
In view of the discussion around slide 32, here is something I should try to better understand:
(here is pdf with more details on this question).
given a simplicial category, what’s the canonical procedure which produces from it a plain category “encoding the same information”.
More concretely:
given a space and a regular epimorphism , we get the simlicial space
Thinking of as a discrete category, this is a simplicial category. But the category I would be after in this simple case is just the Čech groupoid, which is the pair groupoid (codiscrete groupoid) over .
Here its clear what’s going on: the original simplicial set is just the nerve of the Čech groupoid.
But now pick some notion of groupoid of paths in a space . Then we get a genuine simplicial category
I want to form something like the “weak coequalizer”
of this. The concrete description of what I mean by that is in definition 2.11 here. In appendix A.1 of that we describe the universal property of this construction.
This is the groupoid which is generated from paths in and the “jumps between fibers” known from the Čech groupoid, modulo some essentially obvious relations.
Here my question is: what is it we are really doing there? I am thinking that I missed some general nonsense which should, when identified, make all this come out more automatically.
That is essentially the weak coequalizer of
but subject to the constraint that the 2-morphism appearing (due to it being a weak coequalizer) in a sense coequalizes
That last part of the sentence is at best vague. And that’s the reason for my question: what general abstract construction is lurking here in the background?
I had addressed that issue quite a while back originally in an entry called Universal Transition, but back then I didn’t mention the term simplicial category, which is probably necessary to ring a bell here with anyone.
And of course then the next step is to do the same for higher categories. For every notion of strict path -groupoid we get a simplicial -category
and there is the need to transmute this into a mere -category which is with lots of “jumps between fibers” thrown in.
I know how to do this for . Konrad and I are writing this up at the moment. And I think I know, operationally, how to do it for any .
But I don’t yet quite know the best way to think of the general abstract mechanism at work here.
Simlicial n-categories
From p. 4 of
M. Bullejos, E. Faro and V. Blanco, A full and faithful nerve for 2-categories
I suppose that the answer to my question above will involve the Artin-Mazur codiagonal
which is right adjoint to the “total decalage” functor
obtained by pullback along the ordinal sum
I need to see if that really yields the result I am looking for:
for instance, is
equal to the nerve of the the groupoid of definition 2.11?