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The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description:

  1. iso-classes of functors.
  2. formally invert equivalence functors (i.e. localization).

My naive question I: what if we do (1.) and (2.) at the same time?

Basically, we do (1.) at first step, yielding $ho(Gpd)$. Then do (2.), localizing at equivalence (notice: iso-class preserve equivalences).

As Fernando Muro remarked, we obtain nothing new. Since any equivalence functor is a invertible up to natural transformation. This is only true for Sets-theoretic groupoids.

If we consider topological groupoids or Lie groupoids, an equivalent functor (modified internally) may not be invertible up to natural transformation. Therefore, we do get something new.

For topological groupoids or Lie groupoids, perform (1) and (2) we obtain a category which can equally given by calculus of fractions, so the localization functor $ho(Gpd)\to ho(Gpd)(W^{-1})$ preserves limit. Where $ho(Gpd)$ is the category obtained by applying (1). If I am not wrong, the comma groupoid gives a pullback in $ho(Gpd)$.

If we keep natural transformation, only formally invert equivalence functors, this is done by Pronk's bicategory of fractions, denote this bicatgory by $Gpd(W^{-1})$.

This bicategory is closely related, or even equivalent, to stacks.

Question. Is the canonical 2-functor $Gpd\to Gpd(W^{-1})$ preserves 2-limits, in particularly, 2-pullbacks?

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    $\begingroup$ If you do 2 after 1, nothing happens. $\endgroup$
    – Fernando Muro
    Commented Jan 22, 2014 at 20:28
  • $\begingroup$ @FernandoMuro Oh, yes, I see the point here. What confuses me here is that equivalences are quasi-invertable. This may not be true for topological/Lie groupoids. $\endgroup$
    – Ma Ming
    Commented Jan 22, 2014 at 20:37
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    $\begingroup$ You're right, what is hidden here is that all groupoids are fibrant and cofibrant, so the naive homotopy theory (as a quotient by an equivalence relation) coincides with the localization. In general, you can first divide out homotopies and then localize, and in that way the localization has calculus of fractions (at least if you restrict to fibrant or cofibrant objects, which you should do for the homotopy relation to be well defined). $\endgroup$
    – Fernando Muro
    Commented Jan 22, 2014 at 21:07
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    $\begingroup$ Concerning question II, what kind of properties are you interested in? From the usual categorical point of view, homotopy categories are very poor. In general, one can only say that they are 'unstably triangulated' in the sense of Hovey's book. Concerning (co)limits, you just have (co)products. $\endgroup$
    – Fernando Muro
    Commented Jan 23, 2014 at 10:34
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    $\begingroup$ @MaMing The nLab page you cited states that $\operatorname{Ho} \mathbf{Gpd}$ does not have pullbacks. So your claim cannot be correct in general. $\endgroup$
    – Zhen Lin
    Commented Jan 24, 2014 at 16:28

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