Lifting Tensorproductsa Long Non-Adjoint Functors

arXiv: Category Theory, 2016

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

arXiv (Cornell University), 2023

A symmetric tensor category D over an algebraically closed field k is called incompressible if its objects have finite length (D is pretannakian) and every tensor functor out of D is an embedding of a tensor subcategory. E.g., the categories Vec, sVec of vector and supervector spaces are incompressible. Moreover, by Deligne's theorem [De2], if char(k) = 0 then any tensor category of moderate growth uniquely fibres over sVec. This implies that Vec, sVec are the only incompressible categories over k in this class, and perhaps altogether, as we expect that all incompressible categories have moderate growth. Similarly, in characteristic p > 0, we also have incompressible Verlinde categories Verp, Ver + p , and by [CEO] any Frobenius exact category of moderate growth uniquely fibres over Verp, meaning that, in this class, the above categories are the only incompressible ones. More generally, the Verlinde categories Verpn , Ver + p n , n ≤ ∞ introduced in [BEO, Co2] are incompressible, and a key conjecture is that every tensor category of moderate growth uniquely fibres over Verp∞. This would make the above the only incompressible categories in this class (and perhaps altogether). We prove a part of this conjecture, showing that every tensor category of moderate growth fibres over an incompressible one. So it remains to understand incompressible categories, and we prove several results in this direction. Namely, let D-Tann be the category of tensor categories that fibre over D. Then we say that D is subterminal if it is a terminal object of D-Tann (i.e., fibre functors to D are unique when exist), and that D is a Bezrukavnikov category if D-Tann is closed under taking images of tensor functors (=quotient categories). Clearly, a subterminal Bezrukavnikov category is incompressible, and we conjecture that the converse also holds; for instance, tensor subcategories of Verp are known to be subterminal. We prove that furthermore they are Bezrukavnikov categories, generalizing the result of Bezrukavnikov [Be] in the case of Vec, sVec. Finally, we show that a tensor subcategory of a finite incompressible category is incompressible. We also find intrinsic sufficient conditions for incompressibility and subterminality. Namely, D is called maximally nilpotent if the growth rate of the length of the symmetric powers of every object is the minimal one theoretically possible. We show that a maximally nilpotent category is incompressible, and also subterminal if it satisfies an additional geometric reductivity condition (for every morphism X ↠ 1 in D, there exists n > 0 for which Sym n X ↠ 1 is split). Then we verify these conditions for the category Ver2n , thereby showing that it is subterminal.

2006

For a rigid tensor abelian category $T$ over a field $k$ we introduce a notion of a normal quotient $q:T\to Q$. In case $T$ is a Tannaka category, our notion is equivalent to Milne's notion of a normal quotient. More precisely, if $T$ is the category of finite dimensional representations of a groupoid scheme $G$ over $k$, then $Q$ is equivalent to the representation category of a normal subgroupoid scheme of $G$. We describe such a quotient in terms of the subcategory $S$ of $T$ consisting of objects which become trivial in $Q$. We show that, under some condition on $S$, $Q$ is uniquely determined by $S$. If $S$ is an 'etale finite tensor category, we show that the quotient of $T$ by $S$ exists. In particular we show the existence of the base change of $T$ with respect to finite separable field extensions. As an application, we obtain a condition for the exactness of sequences of groupoid schemes in terms of the representation categories.

Journal of Knot Theory and Its Ramifications, 2011

We construct functors from a certain algebraic category Alg, defined by Hopf algebra generators and relations, to the category of vector spaces, based on spherical categories. The category Alg is proposed by Habiro to be isomorphic to the cobordism category of once-punctured surfaces. If the proposal is proved valid, the result of this paper would imply a construction of a TQFT functor based on a spherical category.

International Mathematics Research Notices, 2011

We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of forms. We also discuss the problem of categorification of weak fusion rings, and for the simplest families of such rings, determine which ones are categorifiable.

2020

We introduce semiadditive 2-categories, 2-categories with binary 2-biproducts and a zero object, as a suitable framework for typing tensor calculus. Tensors are the generalization of matrices, whose components have more than two indices.

Journal de l'École polytechnique, 2019

Tensor products and q-characters of HL-modules and monoidal categorifications Tome (2019), p. 581-619.

Cambridge Journal of Mathematics

We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensorideals of certain categories of spectral G-Mackey functors for all finite groups G. Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-toglobal principle and establish that the Balmer-Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson-Iyengar-Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite étale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height 0 and height ∞ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors noting that it bijects onto the height ≤ n chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.

1981

Cahiers de topologie et géométrie différentielle catégoriques, tome 22, n o 3 (1981), p. 267-282 <http://www.numdam.org/item?id=CTGDC_1981__22_3_267_0> © Andrée C. Ehresmann et les auteurs, 1981, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE Vol. XXII-3 (1981) 3e COLLOQUE SUR LES CATEGORIES DEDIE A CHARLES EHRESMANN Amiens, Juillet 1980 Tout cela est fait pour amuser ZEuv dit le Sphinx*).

Journal of Pure and Applied Algebra, 1978