arXiv subjects

Josefien Kuijper

Publications and source records attributed to Josefien Kuijper.

8 recordsLinked to original sources

arXiv2025

Algebraic $K$-theory for squares categories

In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of a squares category to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves.

math.KT↗

arXiv2025

All K-theory is squares K-theory

We show that the K-theory spectra of many assemblers, such as the assembler of polytopes in euclidean, hyperbolic or spherical geometry, as well as the assembler of definable sets, are equivalent to the K-theory spectrum of a squares category. We use this to lift the definable Euler characteristic of definable sets in an o-minimal structure to a map of K-theory spectra.

math.KT↗

arXiv2025

Higher Spherical Scissors Congruence I: Hopf Algebra

In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's reinterpretation of scissors congruence via algebraic K-theory, we lift the Sah algebra to an $(E_\infty, E_1)$-Hopf algebra spectrum whose $π_0$ is the classical Sah algebra. As an application, we show that the reduced spherical scissors congruence $K$-theory groups $\widetilde K_{2n}\big(\mathcal{P}^{S^{2k+1}}_{O(2k+2)}\big)$ are nonzero for all nonnegative integers $n$ and $k$.

math.KT↗

arXiv2025

From Hopf Algebras in Model Categories to Hopf Algebras in $\infty$-categories

We show that algebra objects in model categories can be transferred to algebra objects in $\infty$-categories, without any cofibrancy or fibrancy assumptions on the algebra. We furthermore show under some mild extra assumptions that this correspondence extends to commutative bialgebras and to commutative Hopf algebras.

math.CT↗

arXiv2025

Uniqueness of six-functor formalisms

We present an alternative formulation of Scholze's notions of cohomologically proper and cohomologically étale with respect to an abstract six-functor formalism. These conditions guarantee canonical isomorphisms between the direct and exceptional direct images for certain "proper" morphisms, and between the inverse and exceptional inverse images for certain "étale" morphisms. Using this framework, we prove Scholze's conjecture, showing that a six-functor formalism with sufficiently many cohomologically proper and étale morphisms is uniquely determined by the tensor product and inverse image functors, and can be obtained by a construction of Liu-Zheng and Mann. Additionally, we show that a generalisation of the conjecture fails, and propose a measure of this failure in terms of K-theory.

math.AG↗

arXiv2024

A symmetric monoidal Comparison Lemma

In this note we study symmetric monoidal functors from a symmetric monoidal 1-category to a cartesian symmetric monoidal $\infty$-category, which are in addition hypersheaves for a certain topology. We prove a symmetric monoidal version of the Comparison Lemma, for lax as well as strong symmetric monoidal hypersheaves. For a strong symmetric monoidal functor between symmetric monoidal 1-categories with topologies generated by suitable cd-structures, we show that if the conditions of the Comparison Lemma are satisfied, then there is also an equivalence between categories of lax and strong symmetric monoidal hypersheaves respectively, taking values in a complete cartesian symmetric monoidal $\infty$-category. As an application of this result, we prove a lax symmetric monoidal version of our previous result about hypersheaves that encode compactly supported cohomology theories.

math.CT↗

arXiv2023

An axiomatization of six-functor formalisms

In this paper, we consider some variations on Mann's definition $\infty$-categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthm\"uller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty$-categories, and have recollements. Using Nagata's compactification theorem, we show that Nagata six-functor formalism on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of ``varieties and spans consisting of an open immersion and a proper map''. Using this characterisation, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty$-categories and adjoint triples. We characterise which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty$-categories and adjoint triples, extend to local six-functor formalisms.

math.AG↗

arXiv2023

A descent principle for compactly supported extensions of functors

A characteristic property of cohomology with compact support is the long exact sequence that connects the compactly supported cohomology groups of a space, an open subspace and its complement. Given an arbitrary cohomology theory of algebraic varieties, one can ask whether a compactly supported version exists, satisfying such a long exact sequence. This is the case whenever the cohomology theory satisfies descent for abstract blowups (also known as proper cdh descent). We make this precise by proving an equivalence between certain categories of hypersheaves. We show how several classical and non-trivial results, such as the existence of a unique weight filtration on cohomology with compact support, can be derived from this theorem.

math.AG↗