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?