Bicategories with base change
Published:
Abstract: The notion of double category was first introduced and studied by Ehresmann in [1]&[2]. The structure of a weak double category includes a bicategory of horizontal morphisms and a category of vertical morphisms. So, one can view double categories as a generalization of bicategories in which there are two types of morphisms: vertical morphisms that may be taken to compose strictly among themselves and horizontal morphisms that behave like bimodules. Double categories are also useful for making sense of certain constructions such as calculus of mates, spans, and Parametrized Spectra in homotopy theory.
In this notes I will review fibrant double categories (aka framed bicategories introduced in [3]) in which one has a structure of a bicategory and the base change along vertical morphisms. I will give important example of fibrant double category of bimodules of algebras in which base change corresponds to extension, restriction , and coextension of bi-modules along algebra maps. The idea of a bicategory with base change was first (to my knowledge) formalized in Dominic Veritys PhD thesis [4]. There he uses concept of 2-categories with proarrow equipment which turns out to be equivalent to framed bicategories.
[1] Ehresmann, A. & Ehresmann, C. (1978), ‘Multiple functors. ii. the monoidal closed category of multiple categories’, Cahiers Topologie Geom. Differentielle Vol.19, 295333.
[2] Ehresmann, C. (1963), ‘Categories et structures’, Ann. Sci. Ecole Norm. Sup. Vol.19, 349– 426.
[3] Shulman, M. (2009), ‘Framed bicategories and monoidal fibrations’. URL: arXiv:0706.1286
[4] Verity, D. (1992), ‘Enriched categories, internal categories and change of base (phd thesis)