172 citations · 796 across the 66 of their papers we have counts for
36 papers · 1 filter
Structures on Categories of Polynomials
Brandon T. Shapiro, David I. Spivak
We define the monoidal category of polynomials under composition in any category with finite limits, including both cartesian and vertical morphisms…
All Concepts are
Owen Lynch, Brandon T. Shapiro, David I. Spivak
We show that the double category of comonoids in the category of polynomial functors (previously shown by Ahman-Uustalu and Garner to be equivalent to th…
Duoidal Structures for Compositional Dependence
Brandon T. Shapiro, David I. Spivak
We provide a categorical framework for mathematical objects for which there is both a sort of "independent" and "dependent" composition. Namely we model them as duoidal categories…
Dynamic Operads, Dynamic Categories: From Deep Learning to Prediction Markets
Brandon T. Shapiro, David I. Spivak
Natural organized systems adapt to internal and external pressures and this happens at all levels of the abstraction hierarchy. Wanting to think clearly about this idea motivates o…
A reference for categorical structures on
David I. Spivak
In this document, we collect a list of categorical structures on the category of polynomial functors. There is no implied claim that this list is in any way complet…
Collectives: Compositional protocols for contributions and returns
Nelson Niu, David I. Spivak
We introduce a concept called a collective: an interface with a protocol for aggregating contributions and distributing returns. Through such a protocol, many members may participa…