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Contractible Exact Squares

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Abstract

Exact squares in Cat are not necessarily absolute (i.e., preserved by any 2-functor Cat → Cat), or even preserved by any 2-functor given by exponentiation (−)ℐ : Cat → Cat: if a square is preserved by exponentiation it will be called a contractible exact square. We will characterize diagrammatically these contractible squares, and among them the contractible categories, and the so called fibering and cofibering squares, with especially the comma squares and the adjunction squares. As an application we conclude with a diagrammatical characterization of absolutely absolute Kan extensions and especially of absolutely final functors and of absolutely absolute colimits.

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  1. IMJ Université Paris, Diderot 175 rue du Chevaleret, 75013, Paris, France

    René Guitart

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  1. René Guitart
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Correspondence to René Guitart.

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In homage to my friend George Janelidze, for his 60th birthday

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Guitart, R. Contractible Exact Squares. Appl Categor Struct 22, 873–898 (2014). https://doi.org/10.1007/s10485-013-9353-4

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