arXiv subjects

Niall Taggart

Publications and source records attributed to Niall Taggart.

14 recordsLinked to original sources

arXiv2026

Realification of stably trivial vector bundles

The set of stably trivial complex vector bundles over complex projective spaces and spheres has a natural group structure when the corank is small enough. With respect to this group structure, the operations of taking the underlying real vector bundle (realification) and of adding a trivial line bundle (stabilisation), are group homomorphisms. Building on Hu's recent enumerations of stably trivial complex bundles, we compute these homomorphisms in a range by using Weiss calculus to translate the problem to stable homotopy theory.

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arXiv2026

Deconstructing span categories for profinite groups

One of the major advantages of $\infty$-category theory over classical $1$-category theory is its robust and homotopically meaningful framework for taking (co)limits of diagrams of $\infty$-categories. However, it is both subtle and crucial to specify which variant of the $\infty$-category of $\infty$-categories is being used when forming such (co)limits. In this article, we present a concrete case study illustrating how (co)limits of $\infty$-categories behave in a specific setting. We demonstrate that the span category of a profinite group can be realised as the colimit of the span categories of its quotients by open normal subgroups and provide a number of applications to the world of equivariant (higher) algebra.

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arXiv2025

A Dwyer-Rezk classification for polynomial functors in Weiss calculus

In Goodwillie calculus, unpublished work of Dwyer and Rezk provides a classification of reduced filtered colimit preserving $d$-excisive functors from pointed spaces to spectra as spectrum-valued functors on the category of finite sets of cardinality at most $d$ and epimorphisms. We prove through different methods the analogous result in Weiss calculus: $d$-polynomial functors are equivalent to spectrum-valued functors on the category of finite-dimensional inner product spaces of dimension at most $d$ and orthogonal epimorphisms. Via similar methods we obtain a new proof of the classification of homogeneous functors.

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arXiv2025

On tested Bousfield-Friedlander localizations

We demonstrate that a Bousfield-Friedlander localization with a set of test morphisms in the sense introduced by Bandklayder, Bergner, Griffiths, Johnson, and Santhanam can also be characterized as a left Bousfield localization at the set of test morphisms. This viewpoint enables us to establish a homogeneous model structure associated with any calculus arising from a Bousfield-Friedlander localization of this form. As a corollary, we show that homogeneous functors in discrete calculus coincide up to homotopy with those in Goodwillie calculus. Finally, we illustrate this framework by proving that the polynomial model structure of Weiss calculus is a particular instance of tested Bousfield-Friedlander localization.

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arXiv2025

Symplectic Weiss calculi

We provide two candidates for symplectic Weiss calculus based on two different, but closely related, collections of groups. In the case of the non-compact symplectic groups, i.e., automorphism groups of vector spaces with symplectic forms, we show that the calculus deformation retracts onto unitary calculus as a corollary of the fact that Weiss calculus only depends on the homotopy type of the groupoid core of the diagram category. In the case of the compact symplectic groups, i.e., automorphism groups of quaternion vector spaces, we provide a comparison with the other known versions of Weiss calculus analogous to the comparisons of calculi of the second named author, and classify certain stably trivial quaternion vector bundles over finite cell complexes in a range, using elementary results on convergence of Weiss calculi.

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arXiv2024

An algebraic model for rational excisive functors

We provide a new proof of the rational splitting of excisive endofunctors of spectra as a product of their homogeneous layers independent of rational Tate vanishing. We utilise the analogy between endofunctors of spectra and equivariant stable homotopy theory and as a consequence, we obtain an algebraic model for rational excisive functors.

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arXiv2024

Koszul duality and a classification of stable Weiss towers

We introduce a version of Koszul duality for categories, which extends the Koszul duality of operads and right modules. We demonstrate that the derivatives which appear in Weiss calculus (with values in spectra) form a right module over the Koszul dual of the category of vector spaces and orthogonal surjections, resolving conjectures of Arone--Ching and Espic. Using categorical Fourier transforms, we then classify Weiss towers. In particular, we describe the $n$-th polynomial approximation as a pullback of the $(n-1)$-st polynomial approximation along a ``generalized norm map''.

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arXiv2024

Comparing orthogonal calculus and calculus with Reality

We show that there exists a suitable $C_2$-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus ``up to a shift'' in an analogous way with the recovery of real topological $K$-theory from Atiyah's $K$-theory with Reality via appropriate $C_2$-fixed points.

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arXiv2024

The Goodwillie calculus of polyhedral products

We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. We do this by analysing the behaviour on calculus of the Denham-Suciu fibre sequence, the Iriye-Kishimoto decomposition of the polyhedral product constructed from a collection of pairs of cones and their bases, and the Hilton-Milnor decomposition. As a corollary we show that the Goodwillie calculus of these polyhedral products converges integrally and diverges in $v_h$-periodic homotopy unless the simplicial complex is a full simplex.

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arXiv2023

Recovering unitary calculus from calculus with reality

By analogy with complex $K$--theory and $K$--theory with reality, there are theories of unitary functor calculus and unitary functor calculus with reality, both of which are generalisations of Weiss' orthogonal calculus. In this paper we show that unitary functor calculus can be completely recovered from the unitary functor calculus with reality, in analogy to how complex topological $K$--theory is completely recovered from $K$--theory with reality via forgetting the $C_2$--action.

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arXiv2022

The localization of orthogonal calculus with respect to homology

For a set of maps of based spaces $S$ we construct a version of Weiss' orthogonal calculus which depends only on the $S$-local homotopy type of the functor involved. We show that $S$-local homogeneous functors of degree $n$ are equivalent to levelwise $S$-local spectra with an action of the orthogonal group $O(n)$ via a zigzag of Quillen equivalences between appropriate model categories. Our theory specialises to homological localizations and nullifications at a based space. We give a variety of applications including a reformulation of the Telescope Conjecture in terms of our local orthogonal calculus and a calculus version of Postnikov sections. Our results also apply when considering the orthogonal calculus for functors which take values in spectra.

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arXiv2022

Unitary calculus: model Categories and convergence

We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra - a variation of orthogonal spectra - as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the $n$-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

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arXiv2021

Unitary Functor Calculus with Reality

We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, $BU_\mathbb{R}(-)$. The calculus produces a Taylor tower, the $n$-th layer of which is classified by a spectrum with an action of $C_2 \ltimes U(n)$. We further give model categorical considerations, producing a zig-zag of Quillen equivalences between spectra with an action of $C_2 \ltimes U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the $n$-th layer of the Taylor tower.

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arXiv2021

Comparing the orthogonal and unitary functor calculi

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.

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