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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?

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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.

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  • $\begingroup$ The non-existence of a unique right notion would be unsurprising, given the plethora of notions in e.g. constructive mathematics and even choiceless classical mathematics! $\endgroup$
    – Zhen Lin
    Commented Jul 17, 2013 at 10:26
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The answer to your last question is no. The compact (small) objects in $\infty \mathbf{Grpd}$ are the retracts of finite CW complexes. See, e.g. http://ncatlab.org/nlab/show/compact+object+in+an+%28infinity%2C1%29-category (also google "Wall's finiteness obstruction").

As to whether the first three notions coincide, I would guess yes, but I haven't thought about it much.

EDIT: The third class is strictly larger, as was just pointed out in Mike Shulman's answer. Theorem 2C.5 in Hatcher states that any finite CW complex has the homotopy type of a finite simplicial complex. This shows that the first two notions coincide, and are contained in the third notion (take the poset corresponding to a finite simplicial complex).

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    $\begingroup$ Ha! I added that bit to the nlab earlier today! $\endgroup$
    – Omar Antolín-Camarena
    Commented Jul 16, 2013 at 17:55
  • $\begingroup$ The fact that the compact objects in $\infty \mathbf{Grpd}$ are the retracts of finite CW complexes also appears in Lurie's Higher Topos Theory Remark 5.4.1.6. Note, however, that the finite CW complexes still generate $\infty \mathbf{Grpd}$ under filtered colimits. $\endgroup$
    – Ilan Barnea
    Commented Jul 13, 2015 at 8:55
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Another possible definition of "finite homotopy type" but not from a categorical point of view is suggested by the paper

SPACES WITH FINITELY MANY NON-TRIVIAL HOMOTOPY GROUPS ALL OF WHICH ARE FINITE , GRAHAM ELLIS , Topdogy Vol. 36, No. 2, pp. 501-504, 1997

This shows that for such a connected CW-space, its homotopy type may be represented by a simplicial group such that its group of $ n$ -simplices is finite for each $n \geqslant 0$.

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    $\begingroup$ I've heard those spaces called "π-finite". $\endgroup$
    – Omar Antolín-Camarena
    Commented Jul 16, 2013 at 21:28

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