What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra
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6$\begingroup$ Could you give a reference where such an object is referred to, or elaborate a bit on what you might expect from such a definition? $\endgroup$– Andy SandersCommented Apr 11, 2020 at 20:08
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1$\begingroup$ I think, it should be "homotopy flat" connection. $\endgroup$– Moishe KohanCommented Apr 11, 2020 at 20:50
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1$\begingroup$ One possible definition (which yields an ∞-Riemann–Hilbert correspondence) is in Block and Smith's paper A Riemann--Hilbert correspondence for infinity local systems $\endgroup$– Dmitri PavlovCommented Apr 11, 2020 at 23:54
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$\begingroup$ Andy, It's precisely because I'd like to cite it if it exists, $\endgroup$– Jim StasheffCommented Apr 12, 2020 at 23:55
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1 Answer
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One place where this notion appears (without a formal definition) is in a remark after Corollary 4 in
Cohen, Ralph L.; Stacey, Andrew, Fourier decompositions of Loop bundles, Goerss, Paul (ed.) et al., Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic (K)-theory. Papers from the international conference on algebraic topology, Northwestern University, Evanston, IL, USA, March 24–28, 2002. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 346, 85-95 (2004). ZBL1068.55016.
Remark. The homotopy type of the map of based loop spaces, $\Omega f_\zeta: \Omega M \to U(n)$ can be obtained by taking the holonomy map of a connection on $\zeta$. Therefore this corollary can be interpreted as saying that in order for $L\zeta$ to have a Fourier decomposition, $\zeta$ must admit a “homotopy flat” connection.
Deciphering what they are saying, my guess is that a collection $\nabla$ on a (say) principal $G$-bundle $P\to B$ with simply-connected base $B$ is homotopy flat if the holonomy map $$ \Omega(M)\to G $$ is null-homotopic. (In the case of a flat bundles this map is, of course, constant.) Maybe one of the authors, Andrew Stacey (@user:45) can confirm (or deny) that this is what they meant.
answered Mar 24, 2021 at 16:13
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1$\begingroup$ I always thought that in the flat case, the holonomy map $\Omega M \to G$ should factor as $\Omega M \to \pi_1(M) \to G$, where the first map is obtained by taking $\pi_0$ and the second is a homomorphism. Equivalently, it is locally constant. $\endgroup$ Commented Mar 24, 2021 at 16:51
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1$\begingroup$ @JohnKlein: In the answer I was explicitly assuming (as the authors of the quoted paper) that the base is simply-connected. Otherwise, the right thing to do is to restrict to null-homotopic loops or work in the universal covering space. $\endgroup$ Commented Mar 24, 2021 at 17:29