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It is well known that a topological space is more or less the same as an $\infty$-groupoid. I'm wondering if there is an analogous construction which starts with a manifold endowed with Morse theory (either a function or a gradient-like vector field) and leads to an $(\infty, 1)$-category. The idea should be that morphisms are paths going "downwards".

One can easily come up with at least two definitions, which are obviously non-equivalent: the first one is taking critical points as objects and the space of gradient trajectories as morphisms, and the second one is taking the set of all points as objects and nondecreasing continuous paths as morphisms.

So this boils down to the following: is there any precise sense in which these categories are equivalent? And a minimal question is:

Is there any precise sense in which the two following categories are equivalent - the category with objects $0$ and $1$ and one non-identity morphism from $0$ to $1$, and the category with set of objects $[0, 1]$ and unique morphism between $a$ and $b$ iff $a \leq b$?

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    $\begingroup$ It sounds like you want something like the exit-path infinity-category for the stratification provided by the Morse function. More explicitly, list the critical values in order and identify them with the poset [n] = (0<1<...<n) for some n. Then the Morse function gives an [n]-stratification of the manifold, and every (nice) stratified space has an exit path infty-category: morphisms are paths that at worst go to higher strata (so morphisms within a stratum are invertible). $\endgroup$ – Dylan Wilson Jan 31 '18 at 17:24
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    $\begingroup$ for your 'minimal question': those two infty-categories are wildly different... but if you assert that, say, the objects in [0,1) look roughly the same (so those morphisms are invertible) but the object 1 looks different, so maps to it are not invertible, then you get equivalent infty-categories. (maybe you have to topologize the space of objects though, I dunno...) $\endgroup$ – Dylan Wilson Jan 31 '18 at 17:26
  • $\begingroup$ Thanks for this comment! I think my first definition is efficiently equivalent to you exit path category, and second is equivalent to the exit path category for the filtration ordered by the poset $\mathbb{R}$. Still wonder if these are equivalent in some sense. $\endgroup$ – Lev Soukhanov Jan 31 '18 at 17:39
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    $\begingroup$ Again, in the case of just [0,1] and \{0<1\} as posets, I don't think there's any way in which these are equivalent, unless you make lots of morphisms invertible as I described above. With no extra data, these are just visibly inequivalent posets (the first has uncountably many objects...) I don't see a way to encode the topology of [0,1] without magically making every element equivalent to every other one, which is not the case for the poset 0<1... so I don't think anything like that works. I think you'd have to view [0,1] as stratified by 0<1 and take its exit path infty-category $\endgroup$ – Dylan Wilson Jan 31 '18 at 19:54
  • $\begingroup$ I don't understand why you would want the second version to be equivalent. That uses nothing about Morse theory, the whole point of which I thought was supposed to be to decompose a manifold into a finite collection of big chunks. Why would that be equivalent to cutting it up into uncountably many levels, distinguished by nothing much but being closed? In any case, the only natural way I can see to make $0<1$ equivalent to $[0,1]$ is to invert everything, which you certainly don't want. $\endgroup$ – Kevin Arlin Feb 7 '18 at 5:27
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This problem has a long history going back at least to the paper of Cohen, Jones and Segal (available here: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0883-04.pdf).

In terms of your first definition, the answer is that if $f:M \to \Bbb R$ is a Morse-Smale function on a closed Riemannian manifold, then there is indeed a topological category whose objects are the critical points and where morphisms are given by sequences of contiguous flow lines connecting the critical points.

There are a number of formidable technical difficulties in getting this to work since the composition law needs to be continuous and associative. Many of the papers in this area contain gaps.

However, I would like to unabashedly credit Lizhen Qin (my student) with settling this question. See the papers

http://front.math.ucdavis.edu/1107.5527

http://front.math.ucdavis.edu/1102.2838

http://front.math.ucdavis.edu/1012.3643

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