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

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.

• Thanks to Urs Schreiber explaining some things to me today which led me to consider this question. – Jamie Vicary Apr 8 '15 at 23:40
• – Zhen Lin Apr 9 '15 at 6:22

A CW structure is precisely a presentation of an $\infty$-groupoid, and so "finite CW complex" means precisely "finitely presented $\infty$-groupoid."
• 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... – Qiaochu Yuan Apr 9 '15 at 1:07