Thomason’s homotopy colimit theorem for pseudo-functor

Question

Thomason’s homotopy colimit theorem

R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Camb. Phil. Soc. (1) 85 (1979)91-109

says that for a functor $F:I^{op}\to Cat$, the homotopy colimit and pseudo-colimit are the same. More precisely, there exists a natural homotopy equivalence $$ hocolimt(N F)\to N(\int F), $$ where $\int F$ is the Grothendieck construction (computes pseudo-colimit).

My question is, what if we consider a pseudo-functor $F: I^{op}\to Cat$.

Now the right hand side is still defined. For the left side, he says it is not defined. But if we compose $F$ with nerve, we obtain a pseudo-functor $F: I^{op}\to SSet$, which induces a functor $hoNF: I^{op}\to hoSSet$, then take colimit in $hoSSet$. Are the two sides homotopical equivalent?

Answer 1

The most general result of this form that I know is from 0907.0930: "Classifying spaces for braided monoidal categories and lax diagrams of bicategories" by Carrasco-Cegarra-Garzón. They deal with lax diagrams of bicategories and include (at least) two useful things:

• A rectification functor from lax diagrams to strict diagrams.
• A Thomason theorem of the form you want.