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Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the diagram

$$0 \leftarrow C \rightarrow D$$

in the $\infty$-category of presentable stable $\infty$-categories and continuous functors? For example, is it a localization of $D$?

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Yes. Colimits in $Pr^L$ are the same as limits in $Pr^R$, which are created by the forgetful functor to $Cat_\infty$ (Higher Topos Theory, 5.5.3.18). So the pushout $E$ of your diagram is the pullback, in $Cat_\infty$, of the diagram composed of the right adjoint functors. This means an object in $E$ is an object in $D$ together with an equivalence between its image in $C$ and the zero object, with the obvious mapping spaces. But the space of equivalences between an object and a final object, if not empty, is contractible. So the "extra data" of the equivalence is in fact no extra data at all, so that $E$ is a full subcategory of $D$ (the preimage of the zero object of $C$), and the map $D\to E$ is left adjoint to the inclusion.

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  • $\begingroup$ is E some kind of Verdier quotient of (an appropriate closure of) the image of C in D? $\endgroup$
    – pro
    Commented Jul 4, 2015 at 17:00
  • $\begingroup$ Yes, exactly: you can factor the functor $D\to C$ as a colocalization $D\to C'$ followed by a conservative functor $C'\to C$, and you get the same pushout if you replace $C$ by $C'$. $\endgroup$
    – Marc Hoyois
    Commented Jul 4, 2015 at 17:06
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    $\begingroup$ By the way, the corresponding statement for non-$\infty$ presentable categories (that limits in $Pr^R$ are created by the forgetful functor to $Cat$) is due to Greg Bird (in his thesis, which I believe was from 1976). $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 5, 2015 at 0:38
  • $\begingroup$ 1984, not 1976. $\endgroup$
    – Zippy
    Commented Jul 7, 2015 at 2:32

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