Homotopical categories of logics

Peter Arndt

pp. 13-58

in: Arnold Koslow, Arthur Buchsbaum (eds), The road to universal logic I, Basel, Birkhäuser, 2015

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.