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Kan extensions and the calculus of modules for \(\infty\)-categories. (English) Zbl 1362.18020

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.

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U40 Topological categories, foundations of homotopy theory
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