The free bifibration on a functor
Bryce Clarke
Tallinn University of Technology
Abstract:
Fibrations and opfibrations appear throughout category
theory, and are equivalent to functors into Cat. The construction of
the free (op)fibration on a functor is quite simple, involving the use
of comma categories. A bifibration is a functor which is both a
fibration and an opfibration, and is equivalent to a functor to the
category Adj of categories and adjunctions.
In this talk, I will introduce the construction of the free
bifibration on a functor, which turns out to be a little more involved
than first expected. I will also discuss the close relationship with
double category theory, in particular, freely adding companions and
conjoints to a category. String diagrams for double categories will
make an appearance, and several examples of free bifibrations will be
shown. This talk is based on joint work with Gabriel Scherer and Noam
Zeilberger appearing in our recent preprint:
https://arxiv.org/abs/2511.07314