There are general constructions for homotopy (co)limit of diagrams in spaces $$D:I \rightarrow Top$$
I was wondering if the homotopy (co)limit of discrete diagrams, i.e. when the spaces are all discrete, coincides with (co)limit.
A little update to my question: (in response to Connor)
(i) What kind of diagrams $I$ would be imply they conincide?
(ii) What general theory may I use?
(iii) I am actuallly interested of diagrams indexed by simplex category, $\Delta$.
I have come to an example of interest.
The Segal condition for spaces and sets.
Definition of Segal Space: Let $X_*$ be a simplicial space: the composition of the maps for $n \ge 2$ $$ X_n \rightarrow X_1 \times _{X_0} \cdots \times_{X_0} X_1 \rightarrow X_1 \times^h_{X_0} \cdots \times ^h _{X_0} X_1 $$ is a weak equivalence. The last limit is a homotopy limit.
Definition of Segal Set: Same as above but we require $$ X_n \rightarrow X_1 \times _{X_0} \cdots \times_{X_0} X_1 $$ is a set bijection.
Question: Now suppose we equip each space $X_n$ with discrete topology. $X_*$ a simplicial space equivalent to $X_*$ a simplicial set?
No, the homotopy pushout of $* \leftarrow S^0 \rightarrow *$ is $S^1$ but the pushout as sets is $*$.
If all the maps are injective, then for finite diagrams colimits should agree since all the maps are cofibrations.