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  1. 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 (,2)(\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 (,2)(\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 (,2)(\infty,2)-categories and (2,2)-categories and between (,1)(\infty,1)-categories and (2,1)-categories (so then we could also say something about what happens with weak colimits).

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeDec 1st 2015

    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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeDec 6th 2015

    In principle, it should follow from the ∞-version of “change of base” adjunctions induced by adjunctions between enriching monoidal categories. The adjunction Gpd1Gpd\infty Gpd \rightleftarrows 1Gpd should induce by “change of base” the adjunction (,1)Cat(2,1)Cat(\infty,1)Cat \rightleftarrows (2,1)Cat, and similarly the adjunction (,1)CatCat(\infty,1)Cat \rightleftarrows Cat should induce (,2)Cat2Cat(\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.