2022/07/27 by Femić, Bojana · 1 citation
#Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.13452
We introduce a candidate for the inner hom for Dblstlx, the category of strict double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a pair of lax double functors with four 2-cells resembling distributive laws. We extend this characterization to a 2-category isomorphism q\x\Laxhop(\Aa×\Bb,\Cc) \iso \Laxhop(\Aa, \llbracket\Bb,\Cc\rrbracket). We show that instead of a Gray monoidal product in Dblstlx we obtain a product that in a sense strictifies lax double quasi-functors. We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product, extend it to a 2-functor q\x\Laxhopns(\Aa×\Bb,\Cc)→\Laxhop(\Aa×\Bb,\Cc) and show how it restricts to a biequivalence. The (un)currying 2-functors are studied. We prove that a lax double functor from the trivial double category is a monad in the codomain double category, and show that our above 2-functor in the form q\x\Laxhop(*× *,\Dd)→\Laxhop(*,\Dd) recovers the specification \Comp(\HH(\Dd)):\Mnd\Mnd(\HH(\Dd))→\Mnd(\HH(\Dd)) of the natural transformation \Comp introduced by Street, where \HH(\Dd) is the horizontal 2-category of \Dd.