Brauer–Clifford Group of Lie–Rinehart Algebras

In this paper we define the notion of Brauer–Clifford group for [Formula: see text]-Azumaya algebras when [Formula: see text] is a commutative algebra and[Formula: see text] is a [Formula: see text]-Lie algebra over a commutative ring [Formula: see text]. This is the situation that arises in applications having connections to differential geometry. This Brauer–Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.

Completeness of Graphical Languages for Mixed State Quantum Mechanics

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-calculus, ZW-calculus, and so on. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-SMC of finite-dimensional Hilbert spaces. In recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of how to extend these languages beyond pure quantum mechanics to reason about mixed states and general quantum operations, i.e., completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map that allows one to get rid of a quantum system, an operation that is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction , which transforms any †-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction, we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However, this construction fails for some fringe cases like Clifford+T quantum mechanics, as the category does not have enough isometries.

Some remarks on blueprints and $${\pmb {{\mathbb {F}}}_1}$$-schemes

Over the past two decades several different approaches to defining a geometry over $${{\mathbb F}_1}$$ F 1 have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category $${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$ Sch B ~ of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoid M and a relation on the semiring $$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$ M ⊗ F 1 N , is a monoid object in a certain symmetric monoidal category $${\mathsf B}$$ B , which is shown to be complete, cocomplete, and closed. We prove that every $${\widetilde{{\mathsf B}}}$$ B ~ -scheme $$\Sigma $$ Σ can be associated, through adjunctions, with both a classical scheme $$\Sigma _{\mathbb Z}$$ Σ Z and a scheme $$\underline{\Sigma }$$ Σ ̲ over $${{\mathbb F}_1}$$ F 1 in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhäuser, Boston, 87–100, 2005), together with a natural transformation $$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$ Λ : Σ Z → Σ ̲ ⊗ F 1 Z . Furthermore, as an application, we show that the category of “$${{\mathbb F}_1}$$ F 1 -schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of $${\widetilde{{\mathsf B}}}$$ B ~ -schemes to obtain a larger category, whose objects we call “$${{\mathbb F}_1}$$ F 1 -schemes with relations”.

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

Long Dimodules and Quasitriangular Weak Hopf Monoids

In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is quasitriangular and B coquasitriangular, we also prove that HBLong is braided. As a consequence of this result, we obtain that if H is triangular and B cotriangular, HBLong is an example of a symmetric monoidal category.

Abstract bivariant Cuntz semigroups II

We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative {\mathrm{Cu}}-semirings, and we obtain, under mild conditions, a new characterization of solid {\mathrm{Cu}}-semirings R by the condition that {R\cong\llbracket R,R\rrbracket}.

Abstract Bivariant Cuntz Semigroups

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[\![ S,T ]\!] $ playing the role of morphisms from $S$ to $T$. Applied to $C^{\ast }$-algebras $A$ and $B$, the semigroup $[\![ \operatorname{Cu}(A),\operatorname{Cu}(B) ]\!] $ should be considered as the target in analogs of the universal coefficient theorem for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between $C^{\ast }$-algebras naturally define elements in the respective bivariant Cuntz semigroup.

Integral categories and calculus categories

Differential categories are now an established abstract setting for differentiation. However, not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, sA: !A → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorems is called a calculus category.Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K: !A → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence, we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.

Noetherian schemes over abelian symmetric monoidal categories

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let [Formula: see text] be a commutative monoid object in an abelian symmetric monoidal category [Formula: see text] satisfying certain conditions and let [Formula: see text]. If the subobjects of [Formula: see text] satisfy a certain compactness property, we say that [Formula: see text] is Noetherian. We study the localization of [Formula: see text] with respect to any [Formula: see text] and define the quotient [Formula: see text] of [Formula: see text] with respect to any ideal [Formula: see text]. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over [Formula: see text]. Our notion of a scheme over a symmetric monoidal category [Formula: see text] is that of Toën and Vaquié.