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Jim(JDS that is) has asked me about Cordier’s h.c. nerve construction and pointed me to a recent paper which uses it, and also to a paper by Block-Smith last year on the same subject. This seems to be very related to some of the entries but searches do not give hits. Has anyone looked at this stuff? They use Chen’s iterated integrals etc.
This seems to be very related to some of the entries but searches do not give hits.
What kind of hits are you looking for, exactly? We have for instance a page Lie infinity-algebroid representation.
Has anyone looked at this stuff?
I talked with Camilo and Florian about this in Vienna two months ago. In my words, here is what they are doing:
For $X$ a smooth manifold, write $T X$ for the tangent Lie algebroid. For $V$ a chain complex of vector bundles on $X$, write $end(V)$ for the corresponding endomorphism $\infty$-Lie algebroid.
The Lie integration of $T X$ is the $\infty$-groupoid $\Pi(X)$ of paths in $X$, The Lie integration of $end(V)$ is an $\infty$-groupoid inside the $\infty$-category $Ch_\bullet^\circ$ of $\infty$-vector bundles.
So an $\infty$-representation of $T X$, which is a morphism $\phi : T X \to end(V)$ (a “representation up to homotopy”) must integrate to a morphism of $\infty$-catgeories $\int \phi : \Pi(X) \to Ch_\bullet^\circ$.
This construction $\int$ on such morphisms is what is described in the article. The actual machine that makes this work is an $A_\infty$-algebra homomorphism $\Omega^\bullet(X) \to C^\bullet(X)$. Because both the infintiesimal as well as the finite morphisms here can be expressed in terms of dg-algebra (this is described at Lie infinity-algebroid representation), and its differential forms (= functions on $T X$) for the Lie algebroids and singular cochains (= functions on $Sing X$) for the path groupoid. So that $A_\infty$-algebra homomorphism gives a way of reading in an $end(V)$-valued differential form, computing its parallel transport and producing an $\infty$-functor $\Pi(X) \to Ch_\bullet^\circ$.
Thanks. I had skimmed the papers and that did seem to be what was happening but I am not an expert. I suppose that seeing a reference to Cordier’s paper made me wonder if there was a more subtle link with h.c. stuff. (It all comes down to my feeling that that stuff was ignored for so long and that it is great to see people finally doing the sort of thing that we believed possible back in 1984, when Cordier and I discussed the subject area … on a long very interesting walk near here.:-))
The homotopy coherent nerve comes in when we speak of the $\infty$-category of $\infty$-vector bundles modeled by chain complexes: we start with the $sSet$-enriched category $Ch_\bullet$ of chain complexes and take the homotopy coherent nerve in order to regard it as a quasi-category.
Or some variant of this. In their article Camilo and Florian like to think of the model of $\infty$-categories given by $A_\infty$-categories. They just mention the homotopy-coherent nerve as an alternative route through quasi-categories and don’t dwell on it further.
Homotopy coherent nerve is discussed early in Lurie's book starting with 1.1.5, quite nice presentation of one of the more difficult starting points; it is used much in chaper 2, for example in relation to straightening constructions.
Not sure what you are replying to, but if this is about the general notion of homotoy coherent nerve: the best reference is clearly homotopy coherent nerve :-)
I have added a link in homotopy coherent nerve to the original paper of Cordier.
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