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 $\text{colim}F$ agrees 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!

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!