The n-Category Café

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?

Suppose that about the codiagonal is right. Looks good to me.

Then it’s finally getting pretty close to the full $\infty$-version of what I am talking about for so long now:

Proposed definition: nonabelian differential $\infty$-cocycle aka $\infty$-bundle with connection

a) a smooth space is a sheaf on the category of open subsets of Euclidean space and smooth maps between these;

b) a Lie $\infty$-groupoid is a Kan complex internal to smooth spaces;

c) for $X$ a smooth space, write $$\Pi_\infty(X)$$ for the Lie $\infty$-groupoid whose underlying simplicial smooth space is just the simplicial smooth space $S^\bullet(X)$ of smooth singular simplices in $X$, $S^n(X) = Hom_{smoothSpaces}(\Delta^n,X)$;

d) for $$\pi : Y \to X$$ a regular epimorphism of smooth spaces, write

$$Y^\bullet$$

for the Čech Lie $\infty$-groupoid whose underlying simplicial smooth space is $$\left( \cdots Y^{[3]} \stackrel{\stackrel{\to}{\to}}{\to} Y^{[2]} \stackrel{\to}{\to} Y \right)$$

and write

$$\Pi_\infty^Y(X)$$

for the Lie $\infty$-groupoid whose underlying simplicial smooth space is $$\bar W\left( \Pi_\infty(Y^\bullet) \right) = \bar W \left( \cdots \Pi_\infty(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} \Pi_\infty(Y^{[2]}) \stackrel{\to}{\to} \Pi_\infty(Y) \right)$$

e) assume it is right that this comes, as it should, with canonical morphisms

$$Y^\bullet \hookrightarrow \Pi^Y_\infty(X) \stackrel{\simeq}{\to}\gt \Pi_\infty(X)$$

f) for $g$ any finite dimensional $L_\infty$-algebra, write $$S(\mathrm{W}(g))$$ for the classifying smooth space for $g$-valued forms and $$S(\mathrm{CE}(g))$$ for the classifying smooth space for flat $g$-valued forms;

g) a differential $g$-cocycle on a smooth space $X$ is a choice of regular epimorphism $\pi : Y \to X$ together with a choice of the horizontal morphisms in the diagram

$$\array{ Y^\bullet &\stackrel{g}{\to}& \Pi_\infty(S(CE(g))) & cocycle \\ \downarrow && \downarrow \\ \Pi_\infty^Y(X) &\stackrel{(g,A,F_A)}{\to}& \Pi_\infty(S(\mathrm{W}(g))) & connection data \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{\{P_i(F_A)\}}{\to}& \Pi_\infty(S(\mathrm{W}(g)_{basic})) & characteristic forms }$$

in Lie $\infty$-groupoids;

f) write $$\mathbf{B}^{n} U(1)$$ for the Lie $\infty$-groupoid whose underlying simplicial smooth space is the nerve of the strict globular $n$-groupoid which is trivial except in degree $n$, where it looks like $U(1)$;

and for $\mathbf{B} G$ any one-object Lie $\infty$-groupoid, write

$$\mathbf{B} \mathbf{B} G := \prod_i \mathbf{B}^{n_i} U(1)$$

for $n_i$ the degree of the $i$-th non-trivial rational cohomology group of the realization $B G := |\mathbf{B} G|$,

write

$$\mathbf{E} G$$

for the Lie $\infty$-groupoid whose underlying simplicial smooth space is the image under the décalage functor of $\mathbf{B} G$; this secretly carries a Lie $\infty$-group structure and we write

$$\mathbf{B} \mathbf{E} G$$

for the corresponding one-object Lie $\infty$-groupoid;

then, for $G$ any Lie $\infty$-group, a differential $G$-cocycle on a smooth space $X$ is a choice of regular epimorphism $\pi : Y \to X$ together with a choice of the horizontal morphisms in the diagram

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} G & cocycle \\ \downarrow && \downarrow \\ \Pi_\infty^Y(X) &\stackrel{(g,A,F_A)}{\to}& \mathbf{B}\mathbf{E}G & connection data \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{\{P_i(F_A)\}}{\to}& \mathbf{B} \mathbf{B} G & characteristic forms }$$

in Lie $\infty$-groupoids .

Urs,

You may find that the discussion of the codiagonal and its link with homotopy colimits, stacks, the W construction of simplicial fibre bundles and similar things in Richard Lewis’ at

http://www.informatics.bangor.ac.uk/public/math/research/pgpast.html

may help. He has a nice way of thinking of the simplices which is related to one I have seen recently on some link from this blog. (I can’t recall exactly where but you may identify it more quickly than me.)

I do not think that $\overline{W}$ is a good notation for the codiagonal as that has a slightly different, if related, meaning in simplicial theory, and the codiagonal had already been used quite extensively in lots of places before and the usual notation was $\nabla$. (Balancing $\Delta$ for the diagonal?)