10 citations · 27 across the 12 of their papers we have counts for
12 papers
A Polynomial Construction of Nerves for Higher Categories
Brandon T. Shapiro, David I. Spivak
We show that the construction due to Leinster and Weber of a generalized Lawvere theory for a familially representable monad on a (co)presheaf category, and the associated ``nerve'…
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…
A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks
Rebekah Aduddell, James Fairbanks, Amit Kumar +3
Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, usin…
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…
Enrichment of Algebraic Higher Categories
Brandon Shapiro
We provide a definition of enrichment that applies to a wide variety of categorical structures, generalizing Leinster's theory of enriched -multicategories. As a sample of newly…
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…
Familial Monads as Higher Category Theories
Brandon Shapiro
Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explici…
Additivity and Fiber Sequences for Combinatorial K-Theory
Maru Sarazola, Brandon T. Shapiro
The (A)CGW categories of Campbell and Zakharevich show how finite sets and varieties behave like the objects of an exact category for the purpose of algebraic -theory. These str…
Weak cartesian properties of simplicial sets
Carmen Constantin, Tobias Fritz, Paolo Perrone +1
Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálv…
Partial Evaluations and the Compositional Structure of the Bar Construction
Carmen Constantin, Paolo Perrone, Tobias Fritz +1
The algebraic expression can be evaluated to , but it can also be partially evaluated to . In categorical algebra, such partial evaluations can be defined in…
Densities of Hyperbolic Cusp Invariants
Colin Adams, Rose Kaplan-Kelly, Michael Moore +3
We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with c…