arXiv subjects

Nelson Niu

Publications and source records attributed to Nelson Niu.

4 recordsLinked to original sources

arXiv2026

Maximal compatibility of disklike $G$-transfer systems

Transfer systems are a combinatorial model for $N_{\infty}$-operads, which encode commutative structures in equivariant homotopy theory. Blumberg--Hill and Chan gave criteria for when two transfer systems are a compatible pair, meaning they encode the additive transfers and multiplicative norms of a ring-type structure. In this paper, given a transfer system encoding an additive structure, we give explicit formulae for the maximal transfer system it is compatible with. Our formulae simplify for disklike transfer systems, which typically encode additive structures. Further, we prove that (maximal) compatibility is functorial with respect to the inflation map induced by a quotient of groups, letting us compute maximal compatible transfer systems as inflations of connected transfer systems.

math.AT↗

arXiv2023

Polynomial Functors: A Mathematical Theory of Interaction

This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical theory from the ground up, highlighting pictorical techniques and concrete examples to build intuition and provide applications.

math.CT↗

arXiv2023

Monoidal Structures on Generalized Polynomial Categories

Recently, there has been renewed interest in the theory and applications of de Paiva's dialectica categories and their relationship to the category of polynomial functors. Both fall under the theory of generalized polynomial categories, which are free coproduct completions of free product completions of (monoidal) categories. Here we extend known monoidal structures on polynomial functors and dialectica categories to generalized polynomial categories. We highlight one such monoidal structure, an asymmetric operation generalizing composition of polynomial functors, and show that comonoids with respect to this structure correspond to categories enriched over a related free coproduct completion. Applications include modeling compositional bounds on dynamical systems.

math.CT↗

arXiv2021

Collectives: Compositional protocols for contributions and returns

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 participate in a mutual endeavor. We present a variety of real-world examples of collectives to explore the many kinds of things that members can contribute (such as time, work, ideas, or resources) as well as the many ways these contributions can be aggregated and returns distributed. In addition, we illustrate several ways in which new collectives can be constructed from old, alluding to the fact that all these constructions have a natural mathematical description within the category of polynomial functors equipped with a certain monoidal structure.

math.CT↗