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Categorically Algebraic Foundations for Homotopical Algebra

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Abstract

We investigate a structure for an abstract cylinder endofunctor I whichproduces a good basis for homotopical algebra. It essentially consists ofthe usual operations (faces, degeneracies, connections, symmetries, verticalcomposition) together with a transformation w: I2 → I2, whichwe call lens collapse after its realisation in the standard topologicalcase. This structure, if somewhat heavy, has the interest of being“categorically algebraic”, i.e., based on operations onfunctors. Consequently, it can be naturally lifted from a category A to itscategories of diagrams AS and its slice categories A\X,A/X.Further, the dual structure, based on a cocylinder (or path) endofunctor Pcan be lifted to the category of A-valued sheaves on a site, wheneverthe path functor P preserves limits, and to the category Mon A of internalmonoids, with respect to any monoidal structure of A consistent with P.

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  1. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146, Genova, Italy. e-mail

    Marco Grandis

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  1. Marco Grandis
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Grandis, M. Categorically Algebraic Foundations for Homotopical Algebra. Applied Categorical Structures 5, 363–413 (1997). https://doi.org/10.1023/A:1008620005400

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