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Niles Johnson

Publications and source records attributed to Niles Johnson.

At least 19 recordsLinked to original sources
arXiv2026

Invertibility and parity in symmetric monoidal categories

We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers $\pm$1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.

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arXiv2023

Universal pseudomorphisms, with applications to diagrammatic coherence for braided and symmetric monoidal functors

This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.

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arXiv2022

Homotopy Theory of Enriched Mackey Functors

Mackey functors provide the coefficient systems for equivariant cohomology theories. More generally, enriched presheaf categories provide a classification and organization for many stable model categories of interest. Changing enrichments along $K$-theory multifunctors provides an important tool for constructing spectral Mackey functors from Mackey functors enriched in algebraic structures such as permutative categories. This work gives a detailed development of diagrams, presheaves, and Mackey functors enriched over closed multicategories. Change of enrichment, including the relevant compositionality, is treated with care. This framework is applied to the homotopy theory of enriched diagram and Mackey functor categories, including equivalences of homotopy theories induced by $K$-theory multifunctors. Particular applications of interest include diagrams and Mackey functors enriched in pointed multicategories, permutative categories, and symmetric spectra.

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arXiv2022

Multifunctorial $K$-Theory is an Equivalence of Homotopy Theories

We show that each of the three $K$-theory multifunctors from small permutative categories to $\mathcal{G}_*$-categories, $\mathcal{G}_*$-simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these $K$-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann-Osorno $\mathcal{E}_*$-categories is equivalent to the homotopy theory of pointed simplicial categories.

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arXiv2021

Multicategories Model All Connective Spectra

There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.

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arXiv2021

Multifunctorial Inverse $K$-Theory

We show that Mandell's inverse $K$-theory functor is a categorically-enriched non-symmetric multifunctor. In particular, it preserves algebraic structures parametrized by non-symmetric operads. As applications, we describe how ring categories arise as the images of inverse $K$-theory.

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arXiv2021

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user friendly resource for beginners and experts alike.

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arXiv2020

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.

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arXiv2020

2-Dimensional Categories

This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

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arXiv2019

A bicategorical pasting theorem

We provide an elementary proof of a bicategorical pasting theorem that does not rely on Power's 2-categorical pasting theorem, the bicategorical coherence theorem, or the local characterization of a biequivalence.

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arXiv2019

Quillen's Theorem A and the Whitehead theorem for bicategories

We prove a bicategorical analogue of Quillen's Theorem A. As an application, we deduce the well-known result that a pseudofunctor is a biequivalence if and only if it is essentially surjective on objects, essentially full on 1-cells, and fully faithful on 2-cells.

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arXiv2017

K-theory for 2-categories

We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Γ$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.

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arXiv2017

Extending homotopy theories across adjunctions

Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categories.

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arXiv2017

Stable Postnikov data of Picard 2-categories

Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category $\mathcal{D}$ is an infinite loop space, the zeroth space of the $K$-theory spectrum $K\mathcal{D}$. This spectrum has stable homotopy groups concentrated in levels 0, 1, and 2. In this paper, we describe part of the Postnikov data of $K\mathcal{D}$ in terms of categorical structure. We use this to show that there is no strict skeletal Picard 2-category whose $K$-theory realizes the 2-truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard 2-category $ΣC$ from a Picard 1-category $C$, and show that it commutes with $K$-theory in that $KΣC$ is stably equivalent to $ΣK C$.

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arXiv2014

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, $A_\infty$ spaces, $E_\infty$ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple $T$. In such cases, $T$ is acting on a nice simplicial model category in such a way that $T$ descends to a monad on the homotopy category and defines a category of homotopy $T$-algebras. In this setting there is a forgetful functor from the homotopy category of $T$-algebras to the category of homotopy $T$-algebras. Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy $T$-algebra map to a strict map of $T$-algebras. Once we have a map of $T$-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of $T$-algebra maps and the edge homomorphism on $π_0$ is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the $E_2$-term to be calculable in terms of Quillen cohomology groups. We provide worked examples in $G$-spaces, $G$-spectra, rational $E_\infty$ algebras, and $A_\infty$ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of $E_\infty$ ring spectra to the category of $H_\infty$ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space $E_\infty(Σ^\infty_+ \mathrm{Coker}\, J, L_{K(2)} R)$.

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