Framed Bicategories and Monoidal Fibrations

By Michael Shulman. Theory and Applications of Categories, 2007

223 citations recorded by the source corpus.

In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the `morphisms between 0-cells', such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change `nonalgebraically', using categorical fibrations. The resulting `framed bicategories' assemble into 2-categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a `monoidal fibration', meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/16d297df04d4c5eea45583467c079702518f2bf7

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Citations in the corpus

Polynomial functors and polynomial monads (2009, background): “Having laid the groundwork, our first main result is to assemble polynomial functors into a double category, in fact a framed bicategory in the sense of Shulman [55], hence providing a convenient and precise way of handling the base change operation for polynomial functors.”
Shadows and traces in bicategories (2009, background): “It can also be shown to arise naturally from a natural sort of transformation for a class of “fibrant” double categories; see [35,36].”
Double Categories of Relations (2021, unknown): “This is further supported by the fact that a nice enough 2-topos ends up supporting a Yoneda structure [Weber, 2007] which in any case behaves very much like the proarrows of an equipment [Shulman, 2008].”
Enriched Categories, Internal Categories and Change of Base (2011, methodology): “The double bicategories introduced in section 1.4 for this purpose generalise the double categories used by Shulman, Grandis and Paré, Garner [19], [20] and others by allowing composition to be pseudo-associative in both directions.”
A unified framework for generalized multicategories (2009, unknown): “It turns out that this requires additional structure on our virtual double categories: we need to assume that horizontal arrows can be \restricted" along vertical ones, in a sense made precise in x7. Pseudo double categories with this property were called framed bicategories in [ Shu08 ], where they”