Session S27 - Categories and Topology

Thursday, July 15, 14:00 ~ 14:30 UTC-3

Length 1 double categories via monoidal End-indexings

Juan Orendain

National University of Mexico, UNAM, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Symmetric monoidal structures on framed bicategories descend to symmetric monoidal structures on horizontal bicategories. The axioms defining symmetric monoidal double categories are significantly more tractable than those defining symmetric monoidal bicategories. It is thus convenient to study ways of lifting a given bicategory into a framed bicategory along an appropriate category of vertical morphisms. Solutions to the problem of lifting bicategories to double categories have classically been useful in expressing Kelly and Street's mates correspondence and in proving the 2-dimensional Seifert-van Kampen theorem of Brown et. al., amongst many other applications.

Globularly generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical arrows. Globularly generated double categories form a coreflective sub-2-category of general double categories. This, together with an analysis of the internal structure of globularly generated double categories yields a numerical invariant on general double categories. We call this invariant the length. The length of a double category C measures the complexity of lifting decorated bicategories into C.

It is conjectured that framed bicategories are of length 1. Motivated by this I present a general method for constructing globularly generated double categories of length 1 through extra data in the form of what I will call End-monoidal indexings of decoration categories. The methods presented are related to Moeller and Vasilakopoulou's monoidal Grothendieck construction, to Shulman's construction of framed bicategories from monoidal fibrations on cocartesian categories, and in the case of strict single object and single horizontal morphism 2-groupoids decorated by groups, specialize to semidirect products.

Bibliography

[1] J. Orendain. Internalizing decorated bicategories: The globularily generated condition. Theory and Applications of Categories, Vol.34, 2019, No. 4, pp 80-108.2.

[2]J. Orendain. Free globularily generated double categories. Theory and Applications of Categories, Vol. 34, 2019, No. 42, pp1343-1385.3.

[3] J. Orendain. Free Globularly Generated Double Categories II: The Canonical Double Projection. To appear in Cahiers de topologie et geometrie differentielle categoriques. arXiv:1905.02888

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