The Transportation Of Trifunctors In The Tricategory Of Bicategories

2020-02-04T15:43:27Z (GMT) by Andrew Smith
Tricategories, as the construction for the most general sort of weak 3-category being given by explicit coherence axioms, are a particularly important structure in the study of low-dimensional higher category theory. As such the correct notion of a morphism between tricategories, the Trifunctor, is also an important object of interest. Just as many constructions in mathematics can be realised by using functors between appropriate categories, these constructions can be generalised to the 3-dimensional level by using trifunctors between the appropriate tricategories. Of particular interest are trifunctors into the tricategory of bicategories. Given a mathematical structure laid on top of a base object, it can be useful to transport that structure from the original object to a new object across a suitable sort of equivalence. The collection of trifunctors between two tricategories forms a tricategory of its own. So does the collection of functions from the objects of the source tricategory to the objects of the target tricategory, which form the object level of any trifunctor. Therefore in this case the appropriate notion of equivalence is that of biequivalence, and we would hope to be able to transport the structure of a trifunctor across a collection of biequivalences at the object level. While the transport of structure at lower dimensions is achieved using monadic methods, at the general 3-dimensional level these haven't been developed. This thesis aims to provide a method for transporting the structure of a trifunctor into the tricategory of bicategories across object-indexed biequivalences. We do this by working directly from the definition of trifunctor: by constructing the data needed for the new trifunctor from the data of the original trifunctor and the biequivalences, and then proving that the axioms hold using diagram manipulation techniques.

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