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The Road to Universal Logic pp 13–58Cite as

Birkhäuser

Homotopical Categories of Logics

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Part of the book series: Studies in Universal Logic ((SUL))

Abstract

Categories of logics and translations usually come with a natural notion of when a translation is an equivalence. The datum of a category with a distinguished class of weak equivalences places one into the realm of abstract homotopy theory where notions like homotopy (co)limits and derived functors become available. We analyze some of these notions for categories of logics. We show that, while logics and flexible translations form a badly behaved category with only few (co)limits, they form a well behaved homotopical category which has all homotopy (co)limits. We then outline several natural questions and directions for further research suggested by a homotopy theoretical viewpoint on categories of logics.

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Notes

  1. 1.

    See Example 2.8 for a colimit of flexible morphisms which does exist, but does not behave right.

  2. 2.

    Cubical sets are more common than simplicial sets in directed algebraic topology, but it is also easy to write down a cocubical object in \(\mathcal {G}\hspace {-.4pt}\mathit {enLog}\) to produce a cubical variant of Inf •(L).

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Peter Arndt

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  1. Peter Arndt
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  1. City University of New York, New York, New York, USA

    Arnold Koslow

  2. Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil

    Arthur Buchsbaum

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Arndt, P. (2015). Homotopical Categories of Logics. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10193-4_2

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