Given two diagrams $F_1$ and $F_2$, $C \to Top_{*}$, is there any sufficient condition for the existence of a continuous function on $hocolim(F_1) \to hocolim(F_2)$ where $C$ is a small category. Of course, having a natural transformation between $F_1$ and $F_2$ guarantees the existence of such a map. In my research, I am looking for a map in the twisted setting (https://ncatlab.org/nlab/files/bondalKaprEnhTRiangCat.pdf). So, in stead of a natural transformation, I have multiple maps $q_{w,u}:F_1(u) \to F_2(w)$. I am looking for conditions on $ \{ q_{w,u} \} $ that ensures a continuous map on the respective homotopy colimits. Is there reference in the literature? \par Thank you so much!
2026-03-26 17:51:55.1774547515
conditions for maps between homotopy colimits
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