Symmetric Monoidal Category Research Articles

Jacobson-Morozov Lemma for algebraic supergroups

Given a quasi-reductive algebraic supergroup G, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functorΦx:Rep(G)→Rep(OSp(1|2)) associated to any given element x∈Lie(G)1¯. For nilpotent elements x, we show that the functor Φx can be defined using the Deligne filtration associated to x.We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements x∈Lie(G)1¯ which define an embedding of supergroups OSp(1|2)→G so that x lies in the image of the corresponding Lie algebra homomorphism.

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.

A universal rigid abelian tensor category

We prove that any rigid additive symmetric monoidal category can be mapped to a rigid abelian symmetric monoidal category in a universal way. This yields a novel approach to Grothendieck's standard conjecture D and Voevodsky's smash nilpotence conjecture.

Braided Galois objects and Sweedler cohomology of certain Radford biproducts

We construct the group of H-Galois objects for a flat and cocommutative Hopf algebra in a braided monoidal category with equalizers provided that a certain assumption on the braiding is fulfilled. We show that it is a subgroup of the group of BiGalois objects of Schauenburg, and prove that the latter group is isomorphic to the semidirect product of the group of Hopf automorphisms of H and the group of H-Galois objects. Dropping the assumption on the braiding, we construct the group of H-Galois objects with normal basis. For H cocommutative we construct Sweedler cohomology and prove that the second cohomology group is isomorphic to the group of H-Galois objects with normal basis. We construct the Picard group of invertible H-comodules for a flat and cocommutative Hopf algebra H. We show that every H-Galois object is an invertible H-comodule, yielding a group morphism from the group of H-Galois objects to the Picard group of H. A short exact sequence is constructed relating the second cohomology group and the two latter groups, under the above mentioned assumption on the braiding. We show how our constructions generalize some results for modules over commutative rings, and some other known for symmetric monoidal categories. Examples of Hopf algebras are discussed for which we compute the second cohomology group and the group of Galois objects.

*-Autonomous Envelopes and Conservativity

We prove 2-categorical conservativity for any {0,T}-free fragment of MALL over its corresponding intuitionistic version: that is, that the universal map from a closed symmetric monoidal category to the *-autonomous category that it freely generates is fully faithful, and similarly for other doctrines. This implies that linear logics and graphical calculi for *-autonomous categories can also be interpreted canonically in closed symmetric monoidal categories. In particular, every closed symmetric monoidal category can be fully embedded in a *-autonomous category, preserving both tensor products and internal-homs. In fact, we prove this directly first with a Yoneda-style embedding (an enhanced "Hyland envelope" that can be regarded as a polycategorical form of Day convolution), and deduce 2-conservativity afterwards from Hyland-Schalk double gluing and a technique of Lafont. The same is true for other fragments of *-autonomous structure, such as linear distributivity, and the embedding can be enhanced to preserve any desired family of nonempty limits and colimits.

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.

Reasoning about conscious experience with axiomatic and graphical mathematics

We cast aspects of consciousness in axiomatic mathematical terms, using the graphical calculus of general process theories (a.k.a symmetric monoidal categories and Frobenius algebras therein). This calculus exploits the ontological neutrality of process theories. A toy example using the axiomatic calculus is given to show the power of this approach, recovering other aspects of conscious experience, such as external and internal subjective distinction, privacy or unreadability of personal subjective experience, and phenomenal unity, one of the main issues for scientific studies of consciousness. In fact, these features naturally arise from the compositional nature of axiomatic calculus.

Mapping class group actions from Hopf monoids and ribbon graphs

We show that any pivotal Hopf monoid H in a symmetric monoidal category \mathcal{C} gives rise to actions of mapping class groups of oriented surfaces of genus g\geq 1 with n\geq 1 boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter–Drinfeld modules over H . They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where \mathcal{C} is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter–Drinfeld module structure.

The Kontsevich integral for bottom tangles in handlebodies

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor Z\colon \mathcal{B}\to \hat{A} , where \mathcal{B} is the category of bottom tangles in handlebodies and \hat{A} is the degree-completion of the category \mathbf{A} of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, \mathbf{A} is the linear PROP governing “Casimir Hopf algebras”, which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2 -tensor. The functor Z induces a canonical isomorphism \operatorname{gr}\mathcal{B}\cong\mathbf{A} , where \operatorname{gr}\mathcal{B} is the associated graded of the Vassiliev–Goussarov filtration on \mathcal{B} . To each Drinfeld associator \varphi we associate a ribbon quasi-Hopf algebra H_\varphi in \hat{A} , and we prove that the braided Hopf algebra resulting from H_\varphi by “transmutation” is precisely the image by Z of a canonical Hopf algebra in the braided category \mathcal{B} . Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le–Murakami–Ohtsuki invariant.

Causality in Higher Order Process Theories

Quantum supermaps provide a framework in which higher order quantum processes can act on lower order quantum processes. In doing so, they enable the definition and analysis of new quantum protocols and causal structures. Recently, key features of quantum supermaps were captured through a general categorical framework, which led to a framework of higher order process theories (HOPT). The HOPT framework models lower and higher order transformations in a single unified theory, with its mathematical structure shown to coincide with the notion of a closed symmetric monoidal category. Here we provide an equivalent construction of the HOPT framework from four simple axioms of process-theoretic nature. We then use the HOPT framework to establish connections between foundational features such as causality, determinism and signalling, alongside exploring their interaction with the mathematical structure of *-autonomy.

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} 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}}} 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 Motimes _{{{mathbb F}_1}} {mathbb N}, is a monoid object in a certain symmetric monoidal category {mathsf B}, which is shown to be complete, cocomplete, and closed. We prove that every {widetilde{{mathsf B}}}-scheme Sigma can be associated, through adjunctions, with both a classical scheme Sigma _{mathbb Z} and a scheme underline{Sigma } over {{mathbb 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}. Furthermore, as an application, we show that the category of “{{mathbb 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}}}-schemes to obtain a larger category, whose objects we call “{{mathbb F}_1}-schemes with relations”.

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories

Given any modular category C over an algebraically closed field k, we extract a sequence (Mg)g≥0 of C-bimodules and show that the Hochschild chain complex CH(C;Mg) of C with coefficients in Mg carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus g+1. The ordinary Hochschild complex of C corresponds to CH(C;M0).This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor FC:C-Surfc⟶Chk with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in C. The functor FC satisfies an excision property which is formulated in terms of homotopy coends. In this sense, any modular category gives naturally rise to a modular functor with values in chain complexes. In zeroth homology, it recovers Lyubashenko's mapping class group representations.The chain complexes in our construction are explicitly computable by choosing a marking on the surface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connected and simply connected groupoid of cut systems that appears in the Lego-Teichmüller game by a contractible Kan complex.

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.

A metalanguage for guarded iteration

Notions of guardedness serve to delineate admissible recursive definitions in various settings in a compositional manner. In recent work, we have introduced an axiomatic notion of guardedness in symmetric monoidal categories, which serves as a unifying framework for various examples from program semantics, process algebra, and beyond. In the present paper, we propose a generic metalanguage for guarded iteration based on combining this notion with the fine-grain call-by-value paradigm, which we intend as a unifying programming language for guarded and unguarded iteration in the presence of computational effects. We give a generic (categorical) semantics of this language over a suitable class of strong monads supporting guarded iteration, and show it to be in touch with the standard operational behaviour of iteration by giving a concrete big-step operational semantics for a certain specific instance of the metalanguage and establishing soundness and (computational) adequacy for this case.

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.

Symmetric Monoidal Categories with Attributes

When designing plans in engineering, it is often necessary to consider attributes associated to objects, e.g. the location of a robot. Our aim in this paper is to incorporate attributes into existing categorical formalisms for planning, namely those based on symmetric monoidal categories and string diagrams. To accomplish this, we define a notion of a "symmetric monoidal category with attributes." This is a symmetric monoidal category in which objects are equipped with retrievable information and where the interactions between objects and information are governed by an "attribute structure." We discuss examples and semantics of such categories in the context of robotics to illustrate our definition.

Wiring diagrams as normal forms for computing in symmetric monoidal categories

Applications of category theory often involve symmetric monoidal categories (SMCs), in which abstract processes or operations can be composed in series and parallel. However, in 2020 there remains a dearth of computational tools for working with SMCs. We present an "unbiased" approach to implementing symmetric monoidal categories, based on an operad of directed, acyclic wiring diagrams. Because the interchange law and other laws of a SMC hold identically in a wiring diagram, no rewrite rules are needed to compare diagrams. We discuss the mathematics of the operad of wiring diagrams, as well as its implementation in the software package Catlab.

Compositional Game Theory, Compositionally

We present a new compositional approach to compositional game theory (CGT) based upon Arrows, a concept originally from functional programming, closely related to Tambara modules, and operators to build new Arrows from old. We model equilibria as a bimodule over an Arrow and define an operator to build a new Arrow from such a bimodule over an existing Arrow. We also model strategies as graded Arrows and define an operator which builds a new Arrow by taking the colimit of a graded Arrow. A final operator builds a graded Arrow from a graded bimodule. We use this compositional approach to CGT to show how known and previously unknown variants of open games can be proven to form symmetric monoidal categories.

Dwyer–Kan homotopy theory for cyclic operads

AbstractWe introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.