### 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 *$L_\infty$-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 $L_\infty$-valued forms, see Space and Quantity.

For more on the nature of $\mathbf{E}G := \mathrm{INN}_0(G)$ 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 $n$-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 $X$ and a regular epimorphism $\pi : Y \to X$, we get the simlicial space

$\cdots Y^{[3]} \stackrel{\stackrel{\to}{\to}}{\to} Y^{[2]} \stackrel{\to}{\to} Y \,.$

Thinking of $Y$ 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 $Y$.

Here its clear what’s going on: the original simplicial set is just the nerve of the Čech groupoid.

But now pick some notion $P(X)$ of groupoid of paths in a space $X$. Then we get a genuine simplicial category

$\cdots P(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} P(Y^{[2]}) \stackrel{\to}{\to} P(Y) \,.$

I want to form something like the “weak coequalizer”

$P_1^\pi(X)$

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 $Y$ 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 $P_1^\pi(X)$ is essentially the weak coequalizer of

$P_1(Y^{[2]}) \stackrel{\to}{\to} P_1(Y)$

but subject to the constraint that the 2-morphism appearing (due to it being a weak coequalizer) in a sense coequalizes

$P_1(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} P_1(Y^{[2]}) \,.$

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 $n$-groupoid $P_n(X)$ we get a simplicial $n$-category

$\cdots P_n(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} P_n(Y^{[2]}) \stackrel{\to}{\to} P_n(Y)$

and there is the need to transmute this into a mere $n$-category which is $P_n(Y)$ with lots of “jumps between fibers” thrown in.

I know how to do this for $n=2$. Konrad and I are writing this up at the moment. And I think I know, operationally, how to do it for any $n$.

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$\bar W : BisimplicialSets \to SimplicialSets$

which is right adjoint to the “total decalage” functor

$SimplicialSets \to BisimplicialSets$

obtained by pullback along the ordinal sum

$+ : \Delta \times \Delta \to \Delta \,.$

I need to see if that really yields the result I am looking for:

for instance, is

$\bar W \left( \cdots P_1\left(Y^{[3]}\right) \stackrel{\stackrel{\to}{\to}}{\to} P_1\left(Y^{[2]}\right) \stackrel{\to}{\to} P_1\left(Y\right) \right)$

equal to the nerve of the the groupoid $P_1^Y(X)$ of definition 2.11?