Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’:
- The homotopy type of a simplicial set that has only finitely many non-degenerate simplices.
- The homotopy type of a CW complex that has only finitely many cells.
- The homotopy type of the nerve of a finitely-presentable category.
Homotopy type theory affords another candidate:
- A higher inductive type that admits a finite presentation in some syntactic sense.
Question. Do these notions coincide? To what extent is each one the ‘right’ notion of finiteness for homotopy types? For instance, are these precisely the homotopy types $X$ such that the representable functor $\mathrm{Hom}(X, -) : \infty \mathbf{Grpd} \to \infty \mathbf{Grpd}$ preserves (homotopy) filtered colimits?
Best Answer
I believe the first two notions coincide, but they definitely don't coincide with the third: as was pointed out in a deleted answer, the nerve of any finite group is a (3) but certainly not a (1) or (2) in general. I think most would agree that (3) is not a good notion of finiteness.
As for HITs, any (1) or (2) can be presented by a HIT of a very simple form: all constructors take zero arguments and the source and target of each constructor is an $\infty$-groupoid expression in the previous ones. I haven't written out a proof, but the converse should also be true.
I don't think I would presume to say whether these, or their retracts, are the "right" notion of finiteness. Perhaps there is no unique "right" notion.