## Equipments, double categories, and monoidal fibrations

### Mike Shulman – 23 April 2008

Frequently in mathematics we find objects admitting both "strict morphisms" and "pro-morphisms". Examples include functors and profunctors, functions and spans, diffeomorphisms and cobordisms, and (as recently realized by May & Sigurdsson) continuous maps and parametrized spectra. We will review two descriptions of these situations, one using *equipments* and the other *double categories*, and describe their equivalence. Then we will give several arguments for taking seriously the double-category point of view, including the definition of morphisms and adjunctions, the construction of monoids and enriched categories, and the prospects for generalization. If time permits, we will then describe how many of these situations can be constructed in a uniform way, starting from a monoidal fibration.

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