On the Grothendieck construction for ∞-categories

Abstract

We provide, among other things: (i) a Bousfield-Kan formula for colimits in ∞-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) ∞categorical generalizations of Barwick-Kan's Theorem Bn and Dwyer-Kan-Smith's Theorem Cn (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of diagram ∞-categories.

Parametrized higher category theory

“…The essential content behind such formulas lies in replacing a given diagram C with one fibered over op S that possesses an S -final map to C . As a warmup, we first explain how this goes when S is a point (Corollaries 12.3 and 12.5); the resulting formula appears to be new in the case of coproducts, whereas the case of spaces was first obtained by Aaron Mazel-Gee in [14]. We then apply the S-Bousfield-Kan formula to show that, supposing S op admits multipullbacks, an S-category is S-cocomplete if and only if it admits all S -(co)products and geometric realizations (Corollary 12.15).…”

Adelic descent for K-theory

“…→ Fun(C, E) exists, and is computed pointwisely. More precisely, by [MG,1.16], each H(c) is canonically equivalent to Gr(H)× C c, and this fiber product as ∞-categories is equivalent to the fiber product computed as quasicategories (simplicial sets). Thus [Lur09,4.3.3.10] (with q = id S and δ = p) applies to ensure p !…”

Fibrations of $\infty$-categories

“…Using Quillen's Theorem A, we give a proof of Quillen's Theorem B for 8-categories. See also [Ba1] for a first treatment of Quillen's Theorem B in the context of quasi-categories, and [HM] for a more central treatment, and Theorem 4.23 of [MG2] for a more model-independent treatment. The following is an application of Quillen's Theorem B and the special property of the relative classifying space for left-final and right-initial fibrations, that it is computed fiberwise.…”