#
Framed bicategories and monoidal fibrations

##
Michael Shulman

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.

2000 MSC:
18D05 (Primary), 18D30, 18D10 (Secondary)

*Theory and Applications of Categories,*
Vol. 20, 2008,
No. 18, pp 650-738.

http://www.tac.mta.ca/tac/volumes/20/18/20-18.dvi

http://www.tac.mta.ca/tac/volumes/20/18/20-18.ps

http://www.tac.mta.ca/tac/volumes/20/18/20-18.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/18/20-18.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/18/20-18.ps

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