# Research summary 2018–present

The following is a summary of my research in recent years. My CV has a complete list of my research articles and books, with download links.

## Overview

My research concerns the use of categorical algebra to model stable homotopy theory. In recent years, my research follows two related but distinct branches.

*Modeling stable 2-types*: Continuing my collaboration with Gurski and Osorno, this branch of my research focuses on symmetric monoidal algebra in dimension 2.*Multifunctorial K-theory*: In a new collaboration with Yau, this branch of my research focuses on the properties of multifunctorial K-theory, also known as infinite loop space machinery.

## Modeling stable 2-types

This research branch studies the categorical algebra that corresponds to homotopy-theoretic data in stable dimensions 0, 1, and 2. The existence of such a correspondence is asserted by the Stable Homotopy Hypothesis, a stable analog of Grothendieck's (unstable) Homotopy Hypothesis. Our paper The 2-dimensional stable homotopy hypothesis (2019) proves the assertion in dimension 2. A survey of our work, Topological Invariants from Higher Categories was featured in the September 2019 issue of the AMS Notices.

In 2-categorical opfibrations... (2021) we developed an algebraic group-completion that generalizes Quillen's S⁻¹S construction and proves a 2-dimensional form of Quillen's Theorem B. My graduate student A. Parab is developing an application of this work, using symmetrization of categorical groups to give a computation of K₃(R) for a commutative ring R. This will categorify a classical application that computes K₂(R) via an abelianization of the group GL(R).

As a special case of the 2-dimensional Stable Homotopy Hypothesis, we are currently working to show that invertible chain complexes of Picard 1-categories form a model for the 2-truncation of the sphere spectrum. This would extend the lower-dimensional case, where the 1-truncation of the sphere spectrum is modeled by the 1-category of invertible chain complexes of Abelian groups.

The idea for this project follows from preliminary calculations in our earlier paper Stable Postnikov data of Picard 2-categories (2017, with M. Stephan). Its current status is as follows:

- The basic definitions and properties for chain complexes of Picard categories are developed in the 2019 Ph.D. thesis of my student P.M. Horst.
- A tensor product of permutative categories is developed in our preprint The symmetric monoidal 2-category of permutative categories (under review).
- Extending this tensor product to the (differential) graded case is the subject of current work in progress.

## Multifunctorial K-theory

This research branch focuses on (multi)categorical and homotopical properties of K-theory constructions. The motivating examples are those of classical algebraic K-theory. The general constructions—due to Segal and, later, Elmendorf-Mandell—build connective spectra from permutative categories.

Our monograph Bimonoidal Categories, Eₙ-monoidal Categories, and Algebraic K-Theory, volume III (under review) contains a detailed treatment of the context and construction of these (multi)functors. They are part of an interconnected network of functors between a variety of algebraic and topological contexts. We are interested in two general questions:

- Which of these are categorically enriched multifunctors and, therefore, preserve ring-like structure?
- Which of these are equivalences of homotopy theories, in the sense of Rezk's complete Segal spaces?

Our work addresses these questions as follows.

- In Multifunctorial Inverse K-Theory (2022) we show that Mandell's inverse functor, P, from Γ-categories to permutative categories, is a categorically-enriched multifunctor.
- In Multicategories Model All Connective Spectra (2023) we show that the free functor, F, from multicategories to permutative categories, is an equivalence of homotopy theories. The statement in the paper's title then follows from the modeling result due to Thomason.
- In Homotopy Equivalent Algebraic Structures... (2022) we show that this free F is a categorically-enriched multifunctor.
- In Multifunctorial K-Theory is an Equivalence of Homotopy Theories (2022) we show that the Elmendorf- Mandell construction, and several related intermediate constructions, are equivalences of homotopy theories.

Together, these works give an emerging picture of how to transport ring-like structures between categorical and stable topological contexts. Our monograph Homotopy Theory of Enriched Mackey Functors (under review) synthesizes the above and extends it in two ways. First, we describe multicategorical enrichments of the K-theoretic contexts. Second, we use this to extend the theory to categories of Mackey functors (enriched presheaves) in multicategories and permutative categories. These are of further interest as inputs for equivariant K-theory, and we plan to develop this in future work.