When Do Colimits Agree with Homotopy Colimits? – Algebraic Topology

Question

I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim$(F)=$colim$(F_c)$ where $F_c$ is the cofibrant replacement of $F$. However, it is not always necessary for $F$ to be cofibrant for the colimit and homotopy colimit to be the same. For example, let $\mathcal{C}=b\leftarrow a\rightarrow c$. Then the cofibrant $\mathcal{C}$-diagrams of simplicial sets are ones $Y\leftarrow X\rightarrow Z$ with $X,Y,Z$ cofibrant and the maps $X\to Y$ and $X\to Z$ cofibrations. However, by the left-properness of simplicial sets, I believe we only need one of the maps to be a cofibration for $\text{colim}F$ to agree with $\text{hocolim}F$.

Are there other properties or tools that can say something about the colimit and homotopy colimit of a diagram of simplicial sets being the same?

Thanks!

Answer 1

I don't think we can expect to have one general answer to this question, only a collection of unrelated specialized results. Here are two more:

• In the category of simplicial sets all filtered colimits are homotopy colimits. Added: The gist of the argument can be found in Proposition 1.3 in Quillen's Higher Algebraic K-theory I. I can't seem to find a more precise reference, but I can give a complete alternative argument. Let $J$ be a filtered category. First, observe that a levelwise acyclic Kan fibration of $J$-diagrams induces an acyclic Kan fibration on colimits (since acyclic Kan fibrations are detected by maps out of finite simplicial sets). By K. Brown's Lemma the colimit functor preserves levelwise weak equivalences between $J$-diagrams of Kan complexes. (That functor is not a right Quillen functor but the assumptions of K. Brown's Lemma are really much weaker, see Lemma 1.1.12 in Hovey's Model Categories.) It now follows that the colimit functor preserves weak equivalences between arbitrary $J$-diagrams since there are filtered colimit preserving fibrant replacement functors on the category of simplicial sets, e.g. $\mathrm{Ex}^\infty$. In particular, the colimit of any $J$-diagram is weakly equivalent to the colimit of its projectively cofibrant replacement.
• Splittings of idempotents, which are colimits over the category freely generated by one idempotent, are always homotopy colimits. Added: This follows since splittings of idempotents are retracts and weak equivalences of simplicial sets are closed under retracts.

On the other hand, your example with homotopy pushouts does fit into the general framework of cofibrant diagrams. There may be more than one notion of cofibrant diagram suitable for deriving the colimit functor. In this case, if we denote the objects of the indexing category as $a_0 \leftarrow a_1 \to a_2$ and make it into a Reedy category by declaring the degree of $a_i$ to be $i$, then Reedy cofibrant diagrams are spans where one leg is a cofibration and the colimit functor is a Quillen functor with respect to the resulting Reedy model structure. (Added: See the proof of Lemma 5.2.6 in Hovey's Model Categories for the details.) Perhaps Gregory's example with cubes can also be described in a similar manner, but it is less obvious to me.