Riehl, Emily; Verity, Dominic Kan extensions and the calculus of modules for \(\infty\)-categories. (English) Zbl 1362.18020 Algebr. Geom. Topol. 17, No. 1, 189-271 (2017). Various models of \((\infty,1)\)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an \(\infty\)-cosmos. Roughly speaking, an \(\infty\)-cosmos, as it is used in this paper, looks approximatively like a fibration category enriched over quasi-categories in which all objects are cofibrant and fibrant: the fibrations are called isofibrations and the class of weak equivalences is a class of maps characterized by the enrichment structure (i.e. the mapping spaces). The objects of an \(\infty\)-cosmos are called \(\infty\)-categories and the morphisms between them are called \(\infty\)-functors. In a generic \(\infty\)-cosmos, the authors introduce modules between \(\infty\)-categories by analogy with the \(1\)-category case. The authors develop a general calculus of modules proving that they naturally assemble into a multicategory-like structure called a virtual equipment. Using the calculus of modules, they define and study pointwise Kan extensions which they relate, in the case of cartesian closed \(\infty\)-cosmoi, to limits and colimits of diagrams valued in an \(\infty\)-category. Reviewer: Philippe Gaucher (Paris) Cited in 2 ReviewsCited in 9 Documents MSC: 18G55 Nonabelian homotopical algebra (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology 55U40 Topological categories, foundations of homotopy theory Keywords:categories; modules; profunctors; virtual equipment; pointwise kan extension PDFBibTeX XMLCite \textit{E. Riehl} and \textit{D. Verity}, Algebr. Geom. Topol. 17, No. 1, 189--271 (2017; Zbl 1362.18020) Full Text: DOI arXiv