Hi! Let $I$ denote all the finite subsets of some set (infinite or finite) S. For each n, let $I_n$ be the subset consisting of objects of cardinality n, so that there are no morphisms between the objects in $I_n$. Suppose that we have a diagram $X:I \rightarrow M$ where $M$ denotes some model category (if it is neccessary, assume some nice properties on it too). My question is the following:

Could we present $hocolim X$ as the homotopy colimit of some subdiagram $J \subset I$? Ideally, I would like to be able to find a $J$ that is directed, so that the homotopy colimit will be the mapping telescope. If so, why? Is there any additional properties on X I need for this to be true?

`$I$`

is not directed (unless`$S$`

is finite, but that's a boring case). However, it isdirectand for such categories there is a very explicit procedure for constructing homotopy colimits (of nice enough i.e. Reedy cofibrant diagrams). It proceeds by inductively attaching objects lying over`$I_n$`

and the taking the mapping telescope of the resulting sequence. It is described in detail in Theorem 9.3.5 of arxiv.org/abs/math/0610009v4. – Karol SzumiĆo Mar 28 '13 at 22:43