arXiv subjects

Kaif Hilman

Publications and source records attributed to Kaif Hilman.

9 recordsLinked to original sources

arXiv2026

On the universality of multiexcisive functors

We provide a multiplicative classification of polynomial endofunctors on spectra in terms of their Mackey functors of cross--effects. More precisely, we prove that various categories of multivariable excisive functors from spectra to spectra are symmetric monoidally equivalent to the corresponding variants of spectral Mackey functors. The symmetric monoidal structures appearing here are the Day convolutions on both sides, and the Mackey functors we consider involve variations on the category of finite sets and surjections. The method is first to introduce certain multivariable functors we call subdiagonal functors. By considering them all at once using parametrised category theory, we prove inductively that they all admit Mackey functor descriptions as symmetric monoidal categories, endowing them with a universal property along the way. In particular, specialising this to univariate functors gives a new proof and strengthening of Glasman's result about d-excisive endofunctors on spectra. As application of our perspective, we prove a ``Segal conjecture'' in the context of Goodwillie calculus when d is a prime number.

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arXiv2026

Parametrised noncommutative motives and equivariant cubical descent in algebraic K-theory

For an atomic orbital base category in the sense of Barwick-Dotto-Glasman-Nardin-Shah, we introduce the category of parametrised perfect-stable categories and use it to construct the parametrised version of noncommutative motives in which algebraic K-theory is corepresented. Furthermore, we initiate a rudimentary theory of parametrised cubes which could be of independent interest, generalising some of the elements in Dotto's theory of equivariant Goodwillie calculus beyond the equivariant case. Using this cubical theory, we show that in the equivariant case for finite 2-groups G, the parametrised noncommutative motives canonically refine to G-symmetric monoidal categories. Consequently, this endows the equivariant algebraic K-theory spectra for these groups with the structure of E-infinity-ring spectra equipped with multiplicative norms in the sense of Hill-Hopkins-Ravenel. Along the way, we will also provide a machine to manufacture G-symmetric monoidal categories from symmetric monoidal categories equipped with G-actions and elucidate how the aforementioned parametrised perfect-stable categories relate to Mackey functors valued in perfect-stable categories.

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arXiv2026

Equivariant localizing motives and multiplicative norms on algebraic K-theory

We construct multiplicative norms on equivariant nonconnective algebraic $K$-theory for finite groups $G$. We also construct a genuine equivariant version of THH equipped with a Dennis trace map from K-theory compatible with the multiplicative norms. To do so, we follow the general strategy of Blumberg-Gepner-Tabuada in the nonequivariant case by generalizing their category of localizing motives to the genuine equivariant context, building upon the theory of perfect $G$-stable categories of the first-named author. Crucially, we proceed using the recent perspective on noncommutative motives by the second-named author with Sosnilo and Winges which allows us to deal with non-exact functors on this category of motives. Together with an isotropy separation argument for equivariant cubes, we prove our main theorem that norms of stable categories preserve equivariant motivic equivalences. As an immediate consequence, we obtain a unique equivariant multiplicative refinement of nonconnective algebraic $K$-theory. From these constructions and results, we draw several applications, namely: (1) that the endofunctor of (equivariant) tensor powers on ordinary perfect stable categories preserve motivic equivalences; (2) that the multiplicative norms also preserve the additive motivic equivalences, thus yielding a motivic refinement of a result of Elmanto-Haugseng and Cnossen-Haugseng-Lenz-Linskens that connective algebraic K-theory admits multiplicative norms; (3) we construct a genuine equivariant version of topological Hochschild homology equipped with a Dennis trace map that is compatible with multiplicative norms; and (4) we prove that every genuine $G$-spectrum is the K-theory of a perfect $G$-stable category.

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arXiv2025

Poincaré Duality Pairs of $\infty$-Categories

We introduce a notion of Poincaré duality for pairs of $\infty$-categories, extending Poincaré-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincaré ads of spaces, iterated Poincaré cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of $\infty$-categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincaré triads to all ads. This theory also lays the foundation for future work by the authors on Poincaré cobordism categories, isovariant Poincaré spaces and string topology.

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arXiv2024

Parametrised functor calculus: excision, spheres, and semiadditivity

We lay down the foundations of a theory of parametrised functor calculus, generalising parts of the functor calculus of Goodwillie. We introduce the notion of excisable posets and develop a theory of excisive approximations in this context. As an application, we introduce two different excisable posets when parametrising over an atomic orbital category. By comparing the notions of excisiveness for these two posets, we relate the invertibility of certain spheres with Nardin's notion of parametrised semiadditivity, generalising Wirthmüller's classical result in equivariant homotopy theory for finite groups.

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arXiv2024

Equivariant Poincaré duality for cyclic groups of prime order and the Nielsen realisation problem

In this companion article to [HKK24], we apply the theory of equivariant Poincaré duality developed there in the special case of cyclic groups $C_p$ of prime order to remove, in a special case, a technical condition given by Davis--Lück [DL24] in their work on the Nielsen realisation problem for aspherical manifolds. Along the way, we will also give a complete characterisation of $C_p$--Poincaré spaces as well as introduce a genuine equivariant refinement of the classical notion of virtual Poincaré duality groups which might be of independent interest.

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arXiv2024

Parametrised Poincaré duality and equivariant fixed points methods

In this article, we introduce and develop the notion of parametrised Poincaré duality in the formalism of parametrised higher category theory by Martini-Wolf, in part generalising Cnossen's theory of twisted ambidexterity to the nonpresentable setting. We prove several basechange results, allowing us to move between different coefficient categories and ambient topoi. We then specialise the general framework to yield a good theory of equivariant Poincaré duality spaces for compact Lie groups and apply our basechange results to obtain a suite of isotropy separation methods. Finally, we employ this theory to perform various categorical Smith-theoretic manoeuvres to prove, among other things, a generalisation of a theorem of Atiyah-Bott and Conner-Floyd on group actions with single fixed points.

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arXiv2024

Parametrised Presentability over Orbital Categories

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick-Dotto-Glasman-Nardin-Shah over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in a companion work to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.

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arXiv2024

An equivariant generalisation of McDuff-Segal's group-completion theorem

In this short note, we prove a G-equivariant generalisation of McDuff-Segal's group-completion theorem for finite groups G. A new complication regarding genuine equivariant localisations arises and we resolve this by isolating a simple condition on the homotopy groups of E-infinity-rings in G-spectra. We check that this condition is satisfied when our inputs are a suitable variant of E-infinity-monoids in G-spaces via the existence of multiplicative norm structures, thus giving a localisation formula for their associated G-spherical group rings.

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