Bispans in Quasicategories

A Tambara functor is an algebraic system indexed by the subgroups of a fixed finite group, possessing additive and multiplicative inductions along subgroup inclusions as well as a twisted distributive law between the two inductions.
These arise in equivariant homotopy theory and group representation theory, with examples coming from representation rings, generalized character rings, equivariant \(K\)-theory, Burnside rings, group cohomology, algebraic \(K\)-theory, and homotopy groups of equivariant \(E_{\infty}\)-ring spectra.
Tambara functors are defined using bispan categories, which simultaneously encode the inductive system and distributive law.
Many Tambara functors can be defined in a compatible manner for all groups at once, suggesting the notion of a global Tambara functor.
Encoding the distributivity properties for global equivariant phenomena suggests passage to bispans in the bicategory of finite groupoids, but the complicated nature of bispan composition means that an axiomatic elaboration of global Tambara functors has yet to be provided, although work of Schwede suggests a relationship to global Mackey functors with power operations.
In this thesis, we provide a quasicategorical bispan construction, initiating the development of the higher categorical framework needed to study global Tambara functors.

We provide a construction of a quasicategory of bispans in a locally cartesian closed quasicategory which is compatible with the span quasicategory of Barwick.
In particular, we obtain a quasicategory whose simplices consist of diagrams encoding higher composites of bispans.
To this end, we develop and study quasicategorical analogues of exponential diagrams, which are the categorical construction governing composition in bispan categories.

We first generalize a theory of bispans in the category of finite sets appearing in the thesis of Cranch, creating a ``decomposed'' bispan quasicategory.
The decomposed bispan diagrams of Cranch are sub-simplicial sets of the bispan diagrams used for the main construction, and we establish pleasant properties of these inclusions.
With these results in hand, we prove that what is a priori a simplicial set of bispans is in fact trivially fibered over the decomposed bispan quasicategory, thus obtaining the desired quasicategory of bispans.