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Donald Yau

Publications and source records attributed to Donald Yau.

At least 19 recordsLinked to original sources
arXiv2025

Equivariant Algebraic K-Theories

A cornerstone of algebraic K-theory is the equivalence between the K-theory machines of May, Segal, and Elmendorf and Mandell. Equivariant algebraic K-theory enriches the theory with group actions, making it more powerful and complex. There are a number of equivariant K-theory machines that turn equivariant categorical data into equivariant spectra, the main objects of study in equivariant stable homotopy theory. This work proves that the following four equivariant K-theory machines are appropriately equivalent: Shimakawa equivariant K-theory; the author's enriched multifunctorial equivariant K-theory; the equivariant K-theory of Guillou, May, Merling, and Osorno; and Schwede global equivariant K-theory. Parts 1 and 2 prove the topological equivalence between Shimakawa and multifunctorial equivariant K-theories. Part 3 proves that their categorical parts are equivalent. Part 4 proves that the equivariant K-theory of Guillou, May, Merling, and Osorno is equivalent to Shimakawa K-theory and Schwede global K-theory for each finite group.

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arXiv2024

May's Conjecture on Bimonoidal Functors and Multiplicative Infinite Loop Space Theory

A conjecture of May states that there is an up-to-adjunction strictification of symmetric bimonoidal functors between bipermutative categories. The main result of this paper proves a weaker form of May's conjecture that starts with multiplicatively strong symmetric bimonoidal functors. As the main application, for May's multiplicative infinite loop space machine from bipermutative categories to either E-infinity ring spaces or E-infinity ring spectra, multiplicatively strong symmetric bimonoidal functors can be replaced by strict symmetric bimonoidal functors.

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arXiv2024

Multifunctorial Equivariant Algebraic K-Theory

A central question in equivariant algebraic K-theory asks whether there exists an equivariant K-theory machine from genuine symmetric monoidal G-categories to orthogonal G-spectra that preserves equivariant algebraic structures. We answer this question positively by constructing an enriched multifunctor K from the G-categorically enriched multicategory of O-pseudoalgebras to the symmetric monoidal category of orthogonal G-spectra, for a compact Lie group G and a 1-connected pseudo-commutative G-categorical operad O. As the main application of its enriched multifunctoriality, K preserves all equivariant algebraic structures parametrized by multicategories enriched in either G-spaces or G-categories. For example, for a finite group G and the G-Barratt-Eccles operad, K transports equivariant E-infinity algebras, in the sense of Guillou-May or Blumberg-Hill, of genuine symmetric monoidal G-categories to equivariant E-infinity algebras of orthogonal G-spectra.

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arXiv2023

The Grothendieck Construction of Bipermutative-Indexed Categories and Pseudo Symmetric Inverse K-Theory

The Grothendieck construction is a fundamental link between indexed categories and opfibrations. This work is a detailed study of the Grothendieck construction over a small tight bipermutative category in the context of Cat-enriched multicategories, with applications to inverse K-theory and pseudo symmetric E-infinity-algebras. The ordinary Grothendieck construction over a small category C is a 2-equivalence that sends a C-indexed category to an opfibration over C. We show that the Grothendieck construction over a small tight bipermutative category D is a pseudo symmetric Cat-multifunctor that is generally not a Cat-multifunctor in the symmetric sense. When the projection to D is taken into account, we prove that the Grothendieck construction over D lifts to a non-symmetric Cat-multiequivalence whose codomain is a non-symmetric Cat-multicategory with small permutative opfibrations over D as objects. As applications we show that inverse K-theory, from Gamma-categories to small permutative categories, is a pseudo symmetric Cat-multifunctor but not a Cat-multifunctor in the symmetric sense. It follows that inverse K-theory preserves algebraic structures parametrized by non-symmetric and pseudo symmetric Cat-multifunctors but not Cat-multifunctors in general. As a special case, we observe that inverse K-theory sends pseudo symmetric E-infinity-algebras in Gamma-categories to pseudo symmetric E-infinity-algebras in small permutative categories.

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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-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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arXiv2020

Modular operads and the nerve theorem

We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to André Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.

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arXiv2020

A graphical category for higher modular operads

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $Ω$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from $\mathbf{U}$.

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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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arXiv2019

Infinity Operads and Monoidal Categories with Group Equivariance

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure, and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras. Many illustrations and examples are included. Assuming only basic category theory, this monograph is intended for graduate students and researchers. In addition to being a coherent reference for the topics covered, this book is also suitable for a graduate student seminar and a reading course.

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arXiv2018

Homotopical Adjoint Lifting Theorem

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $Σ$-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter-Shipley about a zig-zag of Quillen equivalences between commutative $H\mathbb{Q}$-algebra spectra and commutative differential graded $\mathbb{Q}$-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization.

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arXiv2018

Higher cyclic operads

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $Ξ$ of trees, which carries a tight relationship to the Moerdijk-Weiss category of rooted trees $Ω$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $Ξ$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.

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