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Foling Zou

Publications and source records attributed to Foling Zou.

18 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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arXiv2026

Real Hochschild homology as an equivariant Loday construction

Equivariant Loday constructions are a means for providing geometric interpretations of equivariant homology theories. They are usually constructed for a simplicial $G$-set and a $G$-Tambara functor. We study situations where -- depending on the isotropy subgroups occurring in the simplicial $G$-set -- one can work with $H$-Tambara functors for a suitable subgroup $H$ of $G$. We apply this to give an interpretation of Real Hochschild homology of discrete $E_\sigma$-rings as equivariant Loday constructions where we consider $2m$-gons with a geometrically defined action of the dihedral groups $D_{2m}$ for all $m \geq 1$. The action of symmetric groups on $1$-skeleta of permutohedra also gives examples with isotropy groups $C_2$.

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arXiv2026

On the Kelly monoidal structure of $\Lambda$-sequences and unital operads

Let $\Lambda$ be the category of based finite sets $\mathbf{n}$ and based injections. We study properties of monoids and modules in $\Lambda$-sequences under the Kelly monoidal structure. In particular, we show that the forgetful functor from right modules in $\Lambda$-sequences to right modules in symmetric sequences is an isomorphism. We show that any compatible lower data extends to a normal oplax monoidal structure and use this to establish a universal normal oplax monoidal structure on $\Lambda$-sequences extending the Kelly product, identifying unital operads to monoids in unital $\Lambda$-sequences for a general symmetric monoidal category $\mathscr{V}$. We also establish a closed monoidal localization theorem.

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arXiv2025

Spoke topological Hochschild homology

Fix primes $p$ and $\ell$, and let $C_p$ be the cyclic group of order $p$. We compute the $C_p$-equivariant spoke topological Hochschild homology of $\underline{\mathbb{F}}_{\ell}$ and prove it exhibits a form of B\"okstedt periodicity. Here spoke topological Hochschild homology is a variant of topological Hochschild homology where one replaces the circle in the construction with the unreduced suspension of $C_p$. As an application, we use this result to give a new proof of the Segal conjecture for the cyclic group of order an odd prime $p$.

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arXiv2025

Equivariant Steenrod Operations

We introduce the notion of $\mathrm{R}$-Eulerian sequences for any $\mathcal{N}_\infty$-ring spectrum $\mathrm{R}$ of finite orientation order. We prove that each $\mathrm{R}$-Eulerian sequence determines a stable $\mathrm{R}$-cohomology operation. Furthermore, we show that the collection of $\mathrm{R}$-Eulerian sequences carries a natural additive and a multiplicative structure which is linear over the coefficient ring. As an application, we specialize to equivariant ordinary cohomology with coefficients in finite fields and construct genuine equivariant Steenrod operations for all finite groups.

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arXiv2025

Equivariant operations in topological Hochschild homology

We observe a new equivariant relationship between topological Hochschild homology and cohomology. We also calculate the topological Hochschild homology of the topological Hochschild cohomology of a finite prime field, which can be viewed as a certain ring of structured operations in this case.

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arXiv2024

The $\mathbb{Z}/p$-equivariant spectrum $BP\mathbb{R}$ for an odd prime $p$

In the present paper, we construct a $\mathbb{Z}/p$-equivariant analog of the $\mathbb{Z}/2$-equivariant spectrum $BP\mathbb{R}$ previously constructed by Hu and Kriz. We prove that this spectrum has some of the properties conjectured by Hill, Hopkins, and Ravenel. Our main construction method is an $\mathbb{Z}/p$-equivariant analog of the Brown-Peterson tower of $BP$, based on a previous description of the $\mathbb{Z}/p$-equivariant Steenrod algebra with constant coefficients by the authors. We also describe several variants of our construction and comparisons with other known equivariant spectra.

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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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arXiv2024

Loday constructions of Tambara functors

Building on work of Hill, Hoyer and Mazur we propose an equivariant version of a Loday construction for $G$-Tambara functors where $G$ is an arbitrary finite group. For any finite simplicial $G$-set and any $G$-Tambara functor, our Loday construction is a simplicial $G$-Tambara functor. We study its properties and examples. For a circle with rotation action by a finite cyclic group our construction agrees with the twisted cyclic nerve of Blumberg, Gerhardt, Hill, and Lawson. We also show how the Loday construction for genuine commutative $G$-ring spectra relates to our algebraic one via the $\underline{\pi}_0$-functor. We describe Real topological Hochschild homology as such a Loday construction.

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arXiv2023

Examples of \'etale extensions of Green functors

We provide new examples of \'etale extensions of Green functors by transferring classical examples of \'etale extensions to the equivariant setting. Our examples are Tambara functors, and we prove Green \'etaleness for them, which implies Tambara \'etaleness. We show that every $C_2$-Galois extensions of fields gives rise to an \'etale extension of $C_2$-Green functors. Here we associate the constant Tambara functor to the base field and the fix-Tambara functor to the extension. We also prove that all $C_n$-Kummer extensions give rise to \'etale extensions for arbitrary finite $n$. \'Etale extensions of fields induce \'etale extension of $G$-Green functors for any finite group $G$ by passing to the corresponding constant $G$-Tambara functors.

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arXiv2023

Equivariant orientation of vector bundles over disconnected base spaces

In this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group $\mathrm{G}$, a precise condition under which an $\mathrm{R}$-orientation of a $\mathrm{G}$-equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel$-$Whitney class of a "homogeneous" $\mathrm{G}$-equivariant bundle with respect to an $\mathbb{E}_\infty^{\mathrm{G}}$-ring spectrum $\mathrm{R}$. As an application, we show that the $2$-fold direct sum of any homogeneous bundle is $\mathrm{H}\underline{\mathcal{A}}_{\mathrm{G}}$-orientable, where $\underline{\mathcal{A}}_{\mathrm{G}}$ is the Burnside Mackey functor. We notice that $\mathrm{H}\underline{\mathcal{A}}_{\mathrm{G}}$-orientability is equivalent to $\mathrm{H}\underline{\mathbb{Z}}$-orientability when the order of $\mathrm{G}$ is odd. When the order of $\mathrm{G}$ is even, we show that a $\mathrm{G}$-equivariant analog of the tautological line bundle over $\mathbb{RP}^\infty$ is $\mathrm{H}\underline{\mathbb{Z}}$-orientable but not $\mathrm{H}\underline{\mathcal{A}}_{\mathrm{G}}$-orientable.

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arXiv2020

A geometric approach to equivariant factorization homology and nonabelian Poincar\'e duality

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an $E_V$-algebra using a two-sided bar construction, generalizing [And10, KM18]. This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincare duality theorem due to several authors. The eNPD states that the scanning map gives a G-equivalence from the equivariant factorization homology to mapping spaces out the one-point compactification of the G-manifolds when the coefficients are G-connected. For non-G-connected coefficients, when the G-manifolds have suitable copies of R in them, the scanning map gives group completions. This generalizes the recognition principle for V -fold loops spaces in [GM17].

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arXiv2020

Notes on equivariant bundles

We compare two notions of $G$-fiber bundles and $G$-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.

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arXiv2020

Equivariant nonabelian Poincar\'e duality and equivariant factorization homology of Thom spectra

In this paper, we study genuine equivariant factorization homology and its interaction with equivariant Thom spectra, which we construct using the language of parametrized higher category theory. We describe the genuine equivariant factorization homology of Thom spectra, and use this description to compute several examples of interest. A key ingredient for our computations is an equivariant nonabelian Poincar\'e duality theorem, in which we prove that factorization homology with coefficients in a $G$-space is given by a mapping space. We compute the Real topological Hochschild homology ($THR$) of the Real bordism spectrum $MU_\mathbb{R}$ and of the equivariant Eilenberg--MacLane spectra $H\underline{\mathbb{F}}_2$ and $H\underline{\mathbb{Z}}_{(2)}$, as well as factorization homology of the sphere $S^{2\sigma}$ with coefficients in these Eilenberg--MacLane spectra. In Appendix B, Jeremy Hahn and Dylan Wilson compute $THR(H\underline{\mathbb{Z}})$.

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arXiv2020

Unital operads, monoids, monads, and bar constructions

We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital $\Lambda$-sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of symmetric sequences. The monads associated to unital operads are the ones of interest in iterated loop space theory and factorization homology, among many other applications. Our new description of unital operads allows an illuminating comparison between the two-sided monadic bar constructions used in such applications and "classical" monoidal two-sided bar constructions. It also allows a more conceptual understanding of the scanning map central to non-abelian Poincar\'e duality in factorization homology.

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arXiv2020

Loday constructions on twisted products and on tori

We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology.

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