Homotopy category
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Construction of and efficient sampling from the simplicial configuration model Open
Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multinode interactions explicitly. With this new description comes the need for prin…
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Grothendieck–Neeman duality and the Wirthmüller isomorphism Open
We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-t…
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Norms in motivic homotopy theory Open
If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable mo…
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The first stable homotopy groups of motivic spheres Open
We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spec…
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A geometric model for the derived category of gentle algebras Open
In this paper we construct a geometric model for the triangulated category generated by the simple modules of any graded gentle algebra. This leads to a geometric model of their perfect derived categories and by a recent paper of Booth, Go…
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Perfection in motivic homotopy theory Open
We prove a topological invariance statement for the Morel-Voevodsky motivic\nhomotopy category, up to inverting exponential characteristics of residue\nfields. This implies in particular that SH[1/p] of characteristic p>0 schemes\nis invar…
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Modalities in homotopy type theory Open
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the t…
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Homotopy theory of homotopy algebras Open
This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphism called infinity-morphism. The m…
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Definability and approximations in triangulated categories Open
We give criteria for subcategories of a compactly generated algebraic\ntriangulated category to be precovering or preenveloping. These criteria are\nformulated in terms of closure conditions involving products, coproducts,\ndirected homoto…
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Reciprocity sheaves Open
We start developing a notion of reciprocity sheaves, generalizing Voevodsky’s homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will…
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Homotopy limits in type theory Open
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoreti…
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Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories Open
We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated cat…
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On the homotopy groups of spheres in homotopy type theory Open
The goal of this thesis is to prove that $π_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, …
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The special fiber of the motivic deformation of the stable homotopy category is algebraic Open
τ -Mod: the abelian category of graded left modules over MU mot * , * /τ .(25) MU mot * , * /τ -Mod 0 : the abelian category of graded left modules over MU mot * , * /τ that are concentrated in Chow-Novikov degree zero.(26) MU mot * , * MU…
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On the equivalence of all models for (∞,2)$(\infty,2)$‐categories Open
The goal of this paper is to provide the last equivalence needed in order to identify all known models for $(\infty,2)$-categories. We do this by showing that Verity's model of saturated $2$-trivial complicial sets is equivalent to Lurie's…
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C2-equivariant stable homotopy from realmotivic stable homotopy Open
We give a method for computing the [math] -equivariant homotopy groups of the Betti realization of a [math] -complete cellular motivic spectrum over [math] in terms of its motivic homotopy groups. More generally, we show that Betti realiza…
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Motivic homotopy theory in derived algebraic geometry Open
In topology, generalized cohomology theories are representable in the stable homotopy category. The analogue in algebraic geometry is the stable motivic homotopy category, constructed by Morel–Voevodsky, where generalized motivic cohomolog…
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Parametrized higher category theory Open
We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G…
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Cdh Descent in Equivariant Homotopy $K$-Theory Open
We construct geometric models for classifying spaces of linear algebraic groups in $G$-equivariant motivic homotopy theory, where $G$ is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the …
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t-Structures on stable derivators and Grothendieck hearts Open
We prove that given any strong, stable derivator and a $t$-structure on its\nbase triangulated category $\\cal D$, the $t$-structure canonically lifts to all\nthe (coherent) diagram categories and each incoherent diagram in the heart\nuniq…
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Symmetric homotopy theory for operads Open
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Bivariant Theories in Motivic Stable Homotopy Open
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of the Grothendieck six functors formalis…
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Discrete topological complexity Open
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex and replacing the concept of homotopy by that of contiguous simplicial maps. We study th…
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Extending Homotopy Type Theory with Strict Equality Open
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is …
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Quantum field theories on categories fibered in groupoids Open
We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with …
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Localizations and completions in motivic homotopy theory Open
Let $K$ be a perfect field and let $E$ be a homotopy commutative ring spectrum in the Morel-Voevodsky stable motivic homotopy category $\mathcal{SH}(K)$. In this work we investigate the relation between the $E$-homology localization and $E…
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Towards a constructive simplicial model of Univalent Foundations Open
We provide a partial solution to the problem of defining a constructive\nversion of Voevodsky's simplicial model of univalent foundations. For this, we\nprove constructive counterparts of the necessary results of simplicial homotopy\ntheor…
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A model for configuration spaces of points Open
We construct a real combinatorial model for the configuration spaces of points of compact smooth oriented manifolds without boundary. We use these models to show that the real homotopy type of configuration spaces of a simply connected suc…
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Waldhausen K-theory of spaces via comodules Open
Let X be a simplicial set. We construct a novel adjunction be- tween the categories RX of retractive spaces over X and ComodX+ of X+- comodules, then apply recent work on left-induced model category structures [5], [16] to establish the ex…
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On locally coherent hearts Open
We show that, under particular conditions, if a t-structure in the unbounded\nderived category of a locally coherent Grothendieck category restricts to the\nbounded derived category of its category of finitely presented objects, then\nits …