10,439 papers in this slice of arXiv.
Philippe Gaucher
We introduce the notion of a reparametrization category with cuts. For every such reparametrization category P, we prove the tensor lemma, namely that the tensor product of two objectwise weak homotopy equivalences of P-spaces is a weak equivalence, and then the left properness of the q-model structure of P
Philippe Gaucher
We prove that the q-model category of P-multipointed d-spaces for P∈{G,M} is not left proper by constructing a weak equivalence whose pushout along a q-cofibration obtained by attaching a single 1-dimensional globular cell is not a weak equivalence. The counterexample is constructed in the category of Δ-Hausdorff Δ-generated spaces. For P=G, all objects are automatically saturated, and for P=M, the original weak equivalence is between saturated objects. The failure is caused by a family of nonconstant execution paths that converges in the ambient mapping space to a constant map which is not an execution path; after the globular cell is attached, this degeneration causes two previously distinct path components to merge.
Ryo Horiuchi
In this note, we show the category of Γ-sets with the sphere spectrum, the smash product, and the substitution product gives rise to an isomix linearly distributive category, which is a categorical semantics of the multiplicative fragment of linear logic. We also show Lydakis' assembly map between these two monoidal products corresponds to the mixor.
Slava Pimenov, Angel Toledo
Recently Batanin and Markl developed a theory of Koszul duality for operads over a general operadic category in the spirit of Ginzburg-Kapranov. However, their approach doesn't apply directly to the operadic category of n-trees, which are used to describe categorical structures. In this paper we present a category of (locally constant) 2-nets which addresses the main difficulty of extending their theory to the case of 2-trees. Specifically, our category of 2-nets is an operadic category containing 2-trees as a full subcategory, and every map between 2-nets admits a factorization into a chain of elementary maps.
Maxim Ivanov
We introduce C-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism f:Γ→G, we associate to every C-lifting functor F a relative quotient Ff(G). This quotient maps epimorphically onto the subgroup determined by F in every f-extension belonging to C. We show that the construction is functorial in f and that, for an f-extension p:G→G, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism f∗H2(Γ;Z)H2(G;Z)⟶kerp∩[G,G]. Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier H2(G,Γ;Z), and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.
Giovanni Ferrer, Brett Hungar, David Penneys +1
In our previous article [arxiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the C∗-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.
Gregory W. Moore, Eliezer Rabinovici, Ranveer Kumar Singh
Given a Lie group G, a level k, and a Lie subgroup H one can construct 2d conformal field theories by either 1.) gauging a nonanomalous H symmetry of the WZW model constructed from (G,k) or 2.) using an algebraic procedure known as the GKO coset construction. The two models are closely related, but not precisely the same: The gauged WZW model is identified with the corresponding GKO model coupled to a 2d topological field theory. The topological theory is characterized by a commutative Frobenius algebra derived from the endomorphisms of an algebra object in a modular tensor category constructed from (G,H,k). The partition function on the torus of the two models differ by a factor of the dimension of this algebra of endomorphisms. Concrete examples are constructed and some applications to string theory and 2d Yang-Mills coupled to nonanomalous matter are briefly discussed. This paper is a summary of a longer companion paper.
Maxime Ramzi
We prove that the ∞-category of localizing motives in the sense of Blumberg--Gepner--Tabuada is not compactly generated, extending a result of Efimov.
Alessandro Linzi
Nakamura--Reyes showed that the category of commutative mosaics---unital reversible hypermagmas, without associativity---is complete, cocomplete, and has free objects, in contrast to commutative polygroups. Krasner's multivalued addition is designed around ultrametric balls; we take that valuation-theoretic motivation as primary and equip mosaics with valuations, as unit-reflecting unitary morphisms into the tropical polygroup T(Γ). Value-preserving morphisms form the slice over T(Γ), which is complete and cocomplete. The larger lax category of valued mosaics over a fixed ordered abelian group Γ is finitely complete and has all small coproducts; it recovers the category of commutative mosaics for the trivial value group, while lax coequalizers for nontrivial Γ remain open. Associativity is analysed via factor nesting, which implies it under totality and characterises it among total product-ultrametric mosaics such as the Krasner hyperfield and T(Γ), yet is strictly weaker without totality. Among total valued mosaics satisfying factor nesting, the Krasner ball axiom yields associativity and upgrades the weak valuation so that sums are ultrametric balls and the superiorly canonical package follows.
Joseph Newton
This paper studies simple Lie algebras in the Verlinde category, with canonical fundamental group action, using combinatorial methods. We give an exhaustive list of such Lie algebras which either have length at most 3, or appear as a subalgebra of the restriction of the Lie algebra of a simple algebraic group along a principal morphism. The results and exceptions answer several questions and conjectures regarding Verlinde categories of algebraic groups and their Lie algebras.
Federico Campanini, Carmelo Antonio Finocchiaro
We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a λ-generated object and the order-theoretic notion of a λ-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.
Xiao-Wu Chen, Jian Liu, Xue-Song Lu +1
In this article, we construct an explicit pre-triangulated category which is not a triangulated category. Its underlying additive category is the category of finitely generated projective modules of the type-A5 preprojective algebra over F2, and the suspension is induced by the graph-reflection automorphism.
Dipankar Maity
We develop a general ∞-categorical framework for internal Eilenberg-MacLane objects, internal homotopy groups, and an internal notion of covering spaces, and study their behavior under suitable localizations. Applying this to the ordinary motivic localization, we identify internal n-Eilenberg-MacLane objects with strongly A1-invariant sheaves of (abelian when n≥2) groups, yielding a formal obstruction to the motivic homotopy category being an ∞-topos. We prove that taking A1-localizations induces an equivalence between classical A1-coverings of a Nisnevich local space and the internal motivic coverings of its motivic localization, providing a streamlined proof of a generalized motivic Van Kampen theorem. Along the way, we establish a Nisnevich-local-to-global A1-connectivity criterion: a k-scheme is A1-connected if and only if it admits a Nisnevich cover by A1-connected schemes such that each pairwise intersection has a (k-)point. Finally, we show that over a general Qcqs base, passing to the birational motivic homotopy category recovers certain essential topos-theoretic properties absent in the ordinary A1-setting. In fact, for a scheme with finitely many generic points, we show that the birational motivic homotopy category is a Postnikov-complete ∞-topos of cohomological dimension 0.
Stefano Ambra, Andrea Montoli, Diana Rodelo
We introduce the notion of R-full Schreier internal category, which is the monoid analogue of the notion of aspherical abelian groupoid. We associate with every R-full Schreier internal category a direction, which is a Schreier split extension with commutative and cancellative kernel. We show that this association is functorial and that such functor is a product preserving cofibration. Thanks to these properties, we equip the connected components of the fibres of such functor with canonical commutative monoid structures. Using the equivalence between Schreier internal categories and crossed semimodules of monoids, we describe these commutative monoids in terms of crossed Schreier extensions.
Arnaud Mayeux
Given a category and a center, that is a collection of pairs (di,Ni) consisting of a morphism di and a sieve Ni over its codomain, the dilatation of is a new category ' in which every n∈Ni factors, uniquely and functorially, through di. This paper presents the theory of dilatations of categories through a full formalization of the construction and its main theorems in the Lean 4 proof assistant, on top of the Mathlib library. An appendix collects a systematic dictionary between the mathematical statements and the Lean declarations that formalize them.
Kevin Coulembier, Jensen O'Sullivan, Daniel Tubbenhauer
This paper studies the asymptotic growth of tensor powers in affine Hecke categories, or equivalently of powers of Kazhdan--Lusztig basis elements in affine Hecke algebras. We prove general bounds in arbitrary affine type, determine precise asymptotics in affine type A1, and establish corresponding results for several natural families in affine type A2.
Changjian Fu, Zhanhong Liang, Ming Lu
For any symmetrizable generalized intersection matrix (GIM) C, we construct an acyclic valued quiver (Q,d) endowed with an involution θ. Let D be the bounded derived category of finite-dimensional representations of (Q,d), and let Σ stand for the suspension functor of D. We show that the orbit category D/(θ∘Σ) carries a canonical triangulated structure and is 2-periodic. Applying Peng--Xiao's construction to this orbit category, we prove that the GIM algebra gim(C) is isomorphic to the integral Ringel--Hall Lie algebra associated with D/(θ∘Σ). As a further application of the above machinery, we investigate elliptic Lie algebras of types D4(1,1), E6(1,1), E7(1,1) and E8(1,1). For each elliptic Dynkin diagram, we define a finite-dimensional algebra A by taking an appropriate quotient of the acyclic quiver Q attached to the GIM matrix C. From the resulting 2-periodic triangulated categories, we build the corresponding Ringel--Hall Lie algebras, and establish a surjective Lie algebra homomorphism from each elliptic Lie algebra to its integral Ringel--Hall counterpart. This map is conjectured to be injective, and its injectivity on real root spaces is confirmed.
Stefano Ambra, Arnaud Duvieusart, Andrea Montoli
We show how the direction functors can be used to develop a cohomology theory for Barr-exact and S-Maltsev categories, where S is a suitable class of split epimorphisms with a fixed section. Using the fact that, for any set B, the category of small categories with B as set of object is S-Maltsev with respect to the class of Schreier points, we show that the cohomology theory of small categories arising from the direction functors coincides with the one introduced by Hoff and Golasinski.
Lukas Müller
Topological defects in quantum field theories are believed to assemble into higher categories with extra structure. This has been made precise for defects in 2- and 3-dimensional oriented topological quantum field theories by Davydov, Kong, and Runkel and by Carqueville, Meusburger, and Schaumann, respectively. In this paper we study the extra structure present on these categories when the topological field theory is additionally reflection positive. We define reflection defect TQFTs as symmetric monoidal functors out of a defect bordism category that intertwine orientation reversal with complex conjugation; they are reflection positive if they satisfy an additional positivity condition. Our main result is that in two dimensions the bicategory of defects TZ associated to a reflection defect TQFT carries the natural structure of an O(2)-dagger bicategory, a structure we define explicitly. If the theory is reflection positive, TZ can be equipped with additional structure closely related to the definition of a 3-Hilbert space (the two agree up to some finiteness and completeness conditions).
Geoff Vooys
In this largely expository paper we provide a deep and explicit exploration and exposition of the tangent structure on the category of schemes Sch/S whose tangent functor T(X)=TX/S is the relative tangent scheme of Grothendieck described in Éléments de Géométrie Algébrique 4. In particular we provide explicit descriptions of the ways that the bifibration of quasicoherent sheaves and bifbration of quesicoherent sheaves of algebras over schemes may be built from the ways in which the bifibrations of modules and commutative algebras over commutative rings interact. We also show the ways in which these interactions give rise to an explicit description of the standard tangent structure on the category of schemes in terms of sheaves of Kähler differentials, properties of the relative spectrum functor, and more. Finally, we show that quasi-coherent sheaves can be reconstructed from their category of differential bundles by showing that for quasi-separated schemes X and Y, there is an isomorphism X≅Y if and only if there is an equivalence of categories DBun(X)≃DBun(Y).