TITLE: What are finite homotopy types?
QUESTION [18 upvotes]: 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?

REPLY [4 votes]: 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 
$n \geqslant  0$.