arXiv subjects

J. Peter May

Publications and source records attributed to J. Peter May.

7 recordsLinked to original sources

arXiv2026

The homotopical monadicity theorem

We give an axiomatic homotopical analog of the classical categorical Beck monadicity theorem. It often holds when classical monadicity fails. This grew out of an understanding of a general context for recognition principles in iterated loop space theory, as treated in the logical sequel ArXiv 2402.03649, but the present result applies differently and more generally. An example gives a new perspective on the old equivalence between simplicial sets and topological spaces: both are equivalent to simplicial topological spaces, and the equivalence implies a curiously close relationship between realizations of simplicial spaces and realizations of their underlying simplicial sets, viewed as discrete simplicial spaces.

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arXiv2026

Orbital presheaves in equivariant infinite loop space theory

Let $G$ be a finite group. Using a new kind of operad, we axiomatize and explore an infinite loop space machine that constructs (genuine) $G$-spectra from suitably structured functors on orbital presheaves, which are just contravariant functors from the orbit category of $G$ to based spaces. The theory leads unexpectedly to a new operadic description of Mackey functors and hence to a definition of ``topological Mackey functors" and a construction of their associated $G$-spectra. It also leads to Picard $G$-spectra, Azumaya ring $G$-spectra, and Brauer $G$-spectra. These constructions raise many unanswered questions.

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arXiv2025

Equivariant infinite loop space theory, the space level story

We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $Ω$-$G$-spectra for any topological group $G$, but our focus is on the construction of genuine $Ω$-$G$-spectra when $G$ is finite. We also show what is and is not true when $G$ is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$ but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.

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arXiv2024

Group completions and the homotopical monadicity theorem

This paper is divided into three parts. In the first part, we give the general abstract axiomatic theory and treat the classical examples of infinite loop space machines with structured spaces or $G$-spaces as input and spectra or $G$-spectra as output. The new prequel paper gives a logically compelling and more general but less useful analog that does not involve group completion. In the second part, we develop a general context of composite adjunctions that feeds into the first. It specializes to give infinite loop space machines that take either orbital presheaves or algebras over categories of operators as input. The new sequel focuses on new constructions and applications when the starting category is that of orbital presheaves of spaces and the output is $G$-spectra. In the brief third part, we show how the multiplicative theory fits into the axiomatic frameworks of the first and second parts.

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arXiv2021

Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem

We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.

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arXiv2019

Symmetric monoidal G-categories and their strictification

We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E_\infty G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine G-spectra as output.

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arXiv2018

A symmetric monoidal and equivariant Segal infinite loop space machine

In [MMO] (arXiv:1704.03413), we reworked and generalized equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category $\scr{F}$ of finite sets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\scr{F}$-$G$-spaces to the symmetric monoidal category of orthogonal $G$-spectra. We relate it multiplicatively to suspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetric monoidal functors from based $G$-spaces and from abelian groups to $\scr{F}$-$G$-spaces. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence is likely to be applicable in other contexts.

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