All about the Grothendieck construction

Abstract

We provide, among other things: (i) a Bousfield--Kan formula for colimits in $\infty$-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) $\infty$-categorical generalizations of Barwick--Kan's Theorem B${}_n$ and Dwyer--Kan--Smith's Theorem C${}_n$ (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 $\infty$-categories.

References in corpus (3)

• Higher Topos Theory
• A user's guide to co/cartesian fibrations
• A Quillen theorem Bn for homotopy pullbacks

Cited by in corpus (8)

• A user's guide to co/cartesian fibrations
• Fibrations of $\infty$-categories
• The colimit of an $\infty$-local system as a twisted tensor product
• Model $\infty$-categories II: Quillen adjunctions
• Goerss--Hopkins obstruction theory for $\infty$-categories
• Model $\infty$-categories III: the fundamental theorem
• Fibrations and Koszul duality in locally Cartesian localisations
• From the Sigma-type to the Grothendieck construction