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There is a very large literature on ‘classical’ 2-category theory largely coming from the Australian Category Theory school. There we have various kinds of weak limits (pseudo limits/bilimits) and, even better, many kinds of weighted colimits.
Now we also have the theory of quasicategories and also $(\infty,2)$-categories. The former, at least, has a well established theory of colimits as well.
I am wondering what tools we have in the literature to compare these notions?
Now obviously there should be a way to take a bicategory and obtain a $(\infty,2)$-category. Also we should be able to take the (2,1) version of a bicategory (all 2-morphisms invertible) and get a quasicategory. Does anyone know how to do this in such a way that it is clear that weak limits in the two situations are preserved? Ideally this would give us models for the adjunction between $(\infty,2)$-categories and (2,2)-categories and between $(\infty,1)$-categories and (2,1)-categories (so then we could also say something about what happens with weak colimits).
For (2, 1)-categories it would seem most convenient to go via Kan-enriched categories. Then the problem would be reduced to the equivalence of various definitions of homotopy limits of diagrams of Kan complexes.
In principle, it should follow from the ∞-version of “change of base” adjunctions induced by adjunctions between enriching monoidal categories. The adjunction $\infty Gpd \rightleftarrows 1Gpd$ should induce by “change of base” the adjunction $(\infty,1)Cat \rightleftarrows (2,1)Cat$, and similarly the adjunction $(\infty,1)Cat \rightleftarrows Cat$ should induce $(\infty,2)Cat \rightleftarrows 2Cat$, and there are general facts about limit-preservation under change of base. But I don’t know whether this sort of change of base has been done in the ∞-context yet.
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