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Bicategories With Base Change

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The notion of double category was first introduced and studied by Ehresmann in 1963 ([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 talk 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 Verity’s PhD thesis [4]. There he uses concept of 2-categories with proarrow equipment which turns out to be equivalent to framed bicategories.

If time permits I will talk about application of fibrant double categories to my research, that is a double categorical formulation of toposes and geometric morphisms which is expected to simplify definition of fibration between toposes.

[1] Andr´ee Ehresmann and Charles Ehresmann. Multiple functors. II. The monoidal closed category of multiple categories. Cahiers Topologie G´eom. Diff´erentielle, 19(3):295–333, 1978.

[2] Charles Ehresmann. Cat´egories structur´ees. Ann. Sci. Ecole Norm. Sup. (3) ´, 80:349–426, 1963.

[3] Mike Shulman, Framed bicategories and monoidal fibrations

[4] Dominic Verity, Enriched Categories, Internal Categories and Change of Base, 1992

[5] Tom Fiore, Double categories and pseudo algebras.

[6] Ronnie Brown and C.B. Spencer, Double groupoids and crossed modules, Cahiers de Topologie et Géométrie Différentielle Catégoriques 17 (1976), 343–362.

This talk is part of the Cargo series.

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