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I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\infty$-categorical equivalences as weak equivalences, are equivalent to topological spaces, with weak equivalences being weak homotopy equivalences.

Now for simplicity, let me focus on the version of the homotopy hypothesis that says that $1$-groupoids, with categorial equivalences as weak equivalences, are equivalent to topological spaces, where weak equivalences are weak $1$-equivalences, ie. maps which induce isomorphisms on $\pi_0\,,\pi_1\,.$

Now on the other hand, Lie's second and third theorems imply that there is an equivalence of categories between simply connected Lie groups and Lie algebras. These theorems generalize to Lie groupoids and Lie algebroids, where simply connected becomes source simply connected (well, in order for Lie's third theorem to hold completely one needs to use smooth spaces which are generalizations of manifolds, but we can always replace "Lie algebroids" with "integrable Lie algebroids", in any case).

Right now these two results may not seem related, for two reasons:

  1. There is no Lie algebroid present on the topological spaces. However, if we work with manifolds instead (or some appropriate class of infinite dimensional manifolds), then $M$ comes with a natural Lie algebroid, namely $TM\,.$ Therefore, there is a canonical Lie algebroid present.

  2. The groupoids in the homotopy hypothesis are discrete, so there is no source simply connected condition. However, the source simply connected integration of $TM$ is the fundamental groupoid $\Pi_1(M)\,,$ and this is Morita equivalent to $\pi_1(M)\,.$ Therefore $\Pi_1(M)\,,$ with the topology associated with the smooth structure, is equivalent to $\Pi_1(M)$ with the discrete topology, and I believe this is the correct groupoid to compare $M$ with in the homotopy hypothesis for 1-types. So in a sense, the source simply connected condition is naturally present.

Now, if we assume that we that we have a notion of weak equivalence of Lie algebroids which implies that a morphism $TM\to TN$ is a weak equivalence if the induced map $M\to N$ is a weak $1$-equivalence, then we seem to get a connection between Lie's theorems and the homotopy hypothesis, ie. the homotopy hypothesis for (smooth) 1-types seems to be implied by a version of Lie's theorems (this wouldn't exactly be Lie's theorems since Lie's theorems use isomorphisms as weak equivalences, but we can use other weak equivalences instead).

I can go into more detail, but has this connection been written about elsewhere in the literature, or is there any reason to doubt this connection?

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    $\begingroup$ Let me point out that the homotopy hypothesis depends on the definition of $\infty$-groupoids. If we take them to be Kan complexes, then it is a theorem (probably due to Quillen), while the hypothesis usually refers to Grothendieck's definition. cf. mathoverflow.net/q/266738 $\endgroup$
    – Z. M
    May 16 at 7:32
  • $\begingroup$ @Z.M Oh I see, thank you for your comment. I was going off of what is stated on nlab. At least the notions agree for homotopy 1-types. $\endgroup$ May 16 at 17:15

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There are analogues of Lie's theorems in homotopy theory, primarily for rational and $p$-adic homotopy types, as well as Lie ∞-groupoids, which can be seen as smooth homotopy types.

In the rational case, this was figured out by Quillen in his 1969 paper Rational homotopy theory, which establishes a Quillen equivalence between the model categories of rational simply connected spaces and rational reduced differential graded Lie algebras.

This was later extended to non-simply connected spaces, see the book Rational Homotopy Theory II by Félix, Halperin, Thomas.

There is also a correspondence in the $p$-adic case, see the survey by Heuts Lie algebra models for unstable homotopy theory.

Finally, one should also mention Lie ∞-integration, which establishes a correspondence between Lie ∞-algebroids and Lie ∞-groupoids.

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  • $\begingroup$ That's interesting, thank you for the references, I wasn't familiar with rational homotopy theory. Though, the thing I want to consider should be closer to a generalization (rather than an analogue) of the fact the homotopy types correspond to $\infty$-groupoids. $\endgroup$ May 17 at 0:37

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