Here’s a question (possibly naive) I have about the smooth infinity topos, which I’ll call $E$. (I guess I could ask this on MathOverflow, but anyone who is likely to know is probably here.)

I’ll write $T$ for infinity-groupoids, and $\Pi\colon E\to T$ for the functor left adjoint to the one sending a space $X$ to the constant simplicial sheaf with value $X$. I’ll identify the category $Man$ of smooth manifolds with a full subcategory of $E$; thus, $\Pi$ sends a manifold to its homotopy type.

Being an infinity topos, $E$ has (derived) internal function objects, which I’ll write as $[X,Y]$. The question is: if $M$ and $N$ are manifolds, is $\Pi[M,N] \approx [\Pi M,\Pi N]$, the latter being the derived mapping space in $T$?

It’s certainly true if $M=\mathbb{R}^k$, since $[\mathbb{R}^k,N]$ is “$\mathbb{R}$-homotopy equivalent” to $N$, and $\Pi$ inverts $\mathbb{R}$-homotopy equivalence.

]]>Hey all,

So I’ve been kind of bugging out trying to find some kind of coherent theory of comonoids in $\infty$-categories. This, for instance, would apply to comonads (as co-associative comonoids in endomorphism categories) among other things. When I try to use Lurie’s stuff, I end up having to trace further and further back to try to prove anything, and end up feeling like I need a theory of cooperads. Somehow comonoid structures seem fundamentally different than monoid structures. Does anyone know how to do this, or if it’s written down clearly anywhere? For instance, Lurie has this nice theorem in one of the DAGs where he shows that monoids are essentially simplicial objects, and this seems to generalize pieces of Emily Riehl’s work with Dominic Verity, except for the fact that there’s no analog for comonoidal objects. It’d be nice to have the analogous statement saying that comonoids are cosimplicial objects in some essential way.

Thanks for any ideas!

-Jon

]]>Rinehart algebras? Is there any treatment of infinity-representations

of sh-Lie-Rinehart algebras? ]]>

My question is: If I get rid of the assumption on n, but still have an abelian group object, etc., then what goes wrong with this classification? That is: In the special case that X corresponds to a space, and A corresponds to an abelian group, what is the difference between an (EM) 0-gerbe banded by A and an A-torsor? Similarly, what is the difference between an (EM) 1-gerbe banded by A and an A-banded gerbe (in the classical setting)?

I tried to parse the nonabelian classification given on the nLab involving a slightly different notion of gerbe in order to apply it to this question, but couldn't manage it.

I suppose another answer to my question would be a clear definition (for all n) of some objects that H^n+1(X, A) does classify. The end of the infty-gerbe article promises this in section 2.3 of something that Urs wrote, but I couldn't find a section 2.3 in the linked page... and clicking on "2." in the description of sections didn't seem to do anything... Maybe something's wrong with my browser. ]]>