1
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ The homotopy category is not cocomplete. It actually has very few colimits apart from coproducts. $\endgroup$ – Fernando Muro Jan 8 '14 at 16:34
  • 2
    $\begingroup$ @MaMing The Grothendieck construction is not a pseudocolimit, but rather a lax (or oplax, depending on taste) colimit. $\endgroup$ – Zhen Lin Jan 9 '14 at 20:02
1
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.