Universal Weil cohomology

The purpose of this work is to highlight the notions of lax braiding and lax for monoidal categories and more generally for promonoidal categories. Lax centres are lax braided. Generally the is a full subcategory of the lax centre, however we show that it is sometimes the case that the two coincide. We identify lax centres of monoidal functor categories in various cases. Introduction Braidings for monoidal categories were introduced in [JS1] and its forerunners. The Z X of a monoidal category X was introduced in [JS0] in the process of proving that the free tortile monoidal category has another universal property. The of a monoidal category is a braided monoidal category. What we now call lax braidings were considered tangentially by Yetter [Yet]. What we now call the lax Z X l of X was considered under the name weak centre by P. Schauenburg [Sch]. The purpose of this work is to highlight the notions of lax braiding and lax for monoidal categories X and more generally for promonoidal categories C. Lax centres turn out to be lax braided monoidal categories. Generally the is a full subcategory of the lax centre, however it is sometimes the case that the two coincide. We have two such theorems under different hypotheses, one i n the case sufficient dual objects exist in the additive context, and the other in the cartesian context. For a promonoidal category C, we relate the lax of the [Day] convolution on C to the convolution on the lax of C. Indeed, sometimes these are equivalent. One reason for being interested in the lax of X is that, if an object X of X is equipped with the structure of monoid in Z X l , then tensoring with X defines a monoidal endofunctor ƒ X of X ; this has applications in cases where the lax can be explicitly identified. 1. Lax braidings for promonoidal categories Let V denote a complete cocomplete symmetric closed monoidal category and let C be a V-enriched category in the sense of [Kel]. A promagmal structure on C consists of two V-functors P op op : C C C V ƒ ƒ ae AE and J : C V ae AE (called the protensor product and prounit). Recall from [Day] that a promonoidal structure on C is a promagmal structure

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\\mathbb F$. Given a tensor category $\\mathcal{C}$, we have two structure invariants of $\\mathcal{C}$: the Green ring (or the representation ring) $r(\\mathcal{C})$ and the Auslander algebra $A(\\mathcal{C})$ of $\\mathcal{C}$. We show that a Krull-Schmit abelian tensor category $\\mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $\\mathcal{C}$. In fact, we can reconstruct the tensor category $\\mathcal{C}$ from its two invariants and the associator system. More general, given a quadruple $(R, A, \\phi, a)$ satisfying certain conditions, where $R$ is a $\\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\\mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $\\phi$ is an algebra map from $A\\otimes_{\\mathbb F}A$ to an algebra $M(R, A)$ constructed from $A$ and $R$, and $a=\\{a_{i,j,l}|1\\leqslant i,j,l\\leqslant n}$ is a family of ``invertible" matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $\\mathcal C$ over $\\mathbb{F}$ such that $R$ is the Green ring of $\\mathcal C$ and $A$ is the Auslander algebra of $\\mathcal C$. In this case, $\\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.

Homotopic Descent over Monoidal Model Categories

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

Quadratic and hermitian forms in additive and abelian categories

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in Agore (Proc Am Math Soc 139:855–863, 2011) on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.

Universal constructions for Hopf algebras

Partial Traces on Additive Categories

Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category \mathcal{E} is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category \mathcal{E} , and can be constructed as the heart \mathcal{LH}(\mathcal{E}) of a t -structure on the bounded derived category \mathbf{D}^b(\mathcal{E}) or as the localization of the category of monomorphisms in \mathcal{E} . However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of \operatorname{LB} -spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category \mathcal{E} can be found as the heart of a \operatorname{t} -structure on the bounded derived category \mathbf{D}^b(\mathcal{E}) , or as the localization of the category of monomorphisms of \mathcal{E} . In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.

This is a “quality of life” article concerning product groups, using the Mizar system [2], [4]. Like a Sonata, this article consists of three movements. The first act, the slowest of the three, builds the infrastructure necessary for the rest of the article. We prove group homomorphisms map arbitrary finite products to arbitrary finite products, introduce a notion of “group yielding” families, as well as families of homomorphisms. We close the first act with defining the inclusion morphism of a subgroup into its parent group, and the projection morphism of a product group onto one of its factors. The second act introduces the universal property of products and its consequences as found in, e.g., Kurosh [7]. Specifically, for the product of an arbitrary family of groups, we prove the center of a product group is the product of centers. More exciting, we prove for a product of a finite family groups, the commutator subgroup of the product is the product of commutator subgroups, but this is because in general: the direct sum of commutator subgroups is the subgroup of the commutator subgroup of the product group, and the commutator subgroup of the product is a subgroup of the product of derived subgroups. We conclude this act by proving a few theorems concerning the image and kernel of morphisms between product groups, as found in Hungerford [5], as well as quotients of product groups. The third act introduces the notion of an internal direct product. Isaacs [6] points out (paraphrasing with Mizar terminology) that the internal direct product is a predicate but the external direct product is a [Mizar] functor. To our delight, we find the bulk of the “recognition theorem” (as stated by Dummit and Foote [3], Aschbacher [1], and Robinson [11]) are already formalized in the heroic work of Nakasho, Okazaki, Yamazaki, and Shimada [9], [8]. We generalize the notion of an internal product to a set of subgroups, proving it is equivalent to the internal product of a family of subgroups [10].

Topological semiinfinite tensor (super)modules

We show that Kazhdan and Lusztig's category of modules for the affine Lie algebra at an admissible level , equivalently the category of finite‐length grading‐restricted generalized modules for the universal affine vertex operator algebra , is a braided tensor category. Although this tensor category is not rigid, we show that the subcategory of all rigid objects in is equal to the subcategory of all projective objects, and that every simple module in has a projective cover. Moreover, we show that the full subcategory of projective objects in is monoidal equivalent to the category of tilting modules for quantum at the root of unity . Using this, we establish a universal property of the tensor category , and as an application, we prove a weak Kazhdan–Lusztig correspondence, that is, we obtain an exact essentially surjective (but not full or faithful) tensor functor from to the category of finite‐dimensional weight modules for the quantum group associated to at the root of unity . We also use the universal property to classify the categories up to (braided) tensor equivalence and to obtain a tensor‐categorical version of quantum Drinfeld–Sokolov reduction, that is, we construct a braided tensor functor from to a category of modules for the Virasoro algebra at central charge .

Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in C_F corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.

Weil cohomologies and derived dg categories

This thesis provides an introductory exploration of the Weil conjectures, focusing on the deep connections between the zeta functions of smooth projective varieties over finite fields and the concept of Weil cohomology. Building on Andr Weils foundational conjectures, we delve into how these conjectures led to the development of cohomology theories capable of capturing arithmetic information about varieties over finite fields. A key result discussed is the Lefschetz trace formula, a powerful tool in cohomology that allows for the computation of point counts of varieties in terms of the traces of an endomorphism on cohomology groups. Through this study, we aim to outline the theoretical framework behind Weil cohomology and its impact on algebraic geometry.