Bastiaan Cnossen
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View article: Universality of Barwick’s Unfurling Construction
Universality of Barwick’s Unfurling Construction Open
Given an $\infty $-category $\mathcal{C}$ with pullbacks, its $(\infty ,2)$-category $\textbf{Span}(\mathcal{C})$ of spans has the universal property of freely adding right adjoints to morphisms in $\mathcal{C}$ satisfying a Beck–Chevalley…
View article: Homotopical commutative rings and bispans
Homotopical commutative rings and bispans Open
We prove that commutative semirings in a cartesian closed presentable ‐category, as defined by Groth, Gepner, and Nikolaus, are equivalent to product‐preserving functors from the (2,1)‐category of bispans of finite sets. In other words, we…
View article: Universality of span 2-categories and the construction of 6-functor formalisms
Universality of span 2-categories and the construction of 6-functor formalisms Open
Given an $\infty$-category $C$ equipped with suitable wide subcategories $I, P \subset E\subset C$, we show that the $(\infty,2)$-category $\text{S}{\scriptstyle\text{PAN}}_2(C,E)_{P,I}$ of higher (or iterated) spans defined by Haugseng ha…
View article: The Adams isomorphism revisited
The Adams isomorphism revisited Open
We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more genera…
View article: Normed equivariant ring spectra and higher Tambara functors
Normed equivariant ring spectra and higher Tambara functors Open
In this paper we extend equivariant infinite loop space theory to take into account multiplicative norms: For every finite group $G$, we construct a multiplicative refinement of the comparison between the $\infty$-categories of connective …
View article: Global spaces and the homotopy theory of stacks
Global spaces and the homotopy theory of stacks Open
We show that the $\infty$-category of global spaces is equivalent to the homotopy localization of the $\infty$-category of sheaves on the site of separated differentiable stacks, following a philosophy proposed by Gepner-Henriques. We furt…
View article: Parametrized (higher) semiadditivity and the universality of spans
Parametrized (higher) semiadditivity and the universality of spans Open
Semiadditivity of an $\infty$-category, i.e. the existence of biproducts, provides it with useful algebraic structure in the form of a canonical enrichment in commutative monoids. This ultimately comes from the fact that the $\infty$-categ…
View article: Homotopical commutative rings and bispans
Homotopical commutative rings and bispans Open
We prove that commutative semirings in a cartesian closed presentable $\infty$-category, as defined by Groth, Gepner, and Nikolaus, are equivalent to product-preserving functors from the $(2,1)$-category of bispans of finite sets. In other…
View article: The Adams isomorphism revisited
The Adams isomorphism revisited Open
We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more genera…
View article: Partial parametrized presentability and the universal property of equivariant spectra
Partial parametrized presentability and the universal property of equivariant spectra Open
We introduce a notion of partial presentability in parametrized higher category theory and investigate its interaction with the concepts of parametrized semiadditivity and stability from arXiv:2301.08240. In particular, we construct the fr…
View article: Parametrized stability and the universal property of global spectra
Parametrized stability and the universal property of global spectra Open
We develop a framework of parametrized semiadditivity and stability with respect to so-called atomic orbital subcategories of an indexing $\infty$-category $T$, extending work of Nardin. Specializing this framework, we introduce global $\i…