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An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions.

Is the infinity-groupoid of a finite CW complex finitely-presented?

It seems to me this question is relevant for homotopy type theory, in which topological spaces are constructed as higher inductive types from a finite family of generators.

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  • $\begingroup$ Thanks to Urs Schreiber explaining some things to me today which led me to consider this question. $\endgroup$ – Jamie Vicary Apr 8 '15 at 23:40
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    $\begingroup$ See Mike Shulman's answer. $\endgroup$ – Zhen Lin Apr 9 '15 at 6:22
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A CW structure is precisely a presentation of an $\infty$-groupoid, and so "finite CW complex" means precisely "finitely presented $\infty$-groupoid."

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  • $\begingroup$ thanks. This makes sense to me, although I do not quite see why it is a trivial statement... I will think further. $\endgroup$ – Jamie Vicary Apr 9 '15 at 0:22
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    $\begingroup$ What model of an oo-groupoid are you using? Clearly n-cells are generators in dimension n, but how does one interpret the attaching maps? Split in half and let the halves be source and target in a globular oo-groupoid? $\endgroup$ – David Roberts Apr 9 '15 at 0:50
  • $\begingroup$ This is a model-independent statement. Of course it depends on what the OP means by "free $\infty$-groupoid on a finite family of generators," but I think this is a reasonable interpretation. Think first about how one presents a group $G$ in terms of generators and relations. Topologically generators correspond to $1$-cells and relations correspond to $2$-cells. So in fact a presentation describes a CW complex of dimension $2$ (the presentation complex). This is the beginning of a presentation / CW description of $BG$, and one now needs to add $3$-cells, etc. These are simultaneously... $\endgroup$ – Qiaochu Yuan Apr 9 '15 at 1:07
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    $\begingroup$ ...new "generators" in the sense that they may (before we kill them again) give rise to higher homotopy and new "relations" in the sense that they themselves kill homotopy. $\endgroup$ – Qiaochu Yuan Apr 9 '15 at 1:08
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    $\begingroup$ Another way to say it: a presentation of a group is a description of that group as the cokernel of a map between free groups. Keeping in mind the analogy to free resolutions of modules, we might say more generally that "presentation" means "description of an object as a colimit of free objects." And a CW decomposition is precisely a description of a space as an iterated homotopy pushout of spheres (which themselves are iterated homotopy pushouts of points). $\endgroup$ – Qiaochu Yuan Apr 9 '15 at 1:14

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