Brian Day
*-Autonomous categories were initially defined by M. Barr to describe a type of duality carried by many monoidal closed categories. Later they were generalised by the current author to include *-autonomous promonoidal categories. Together, these structures under "convolution" product give a clear indication of the usefulness of *-autonomy in quantum mathematics and related areas.
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Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous categories.
We discuss cyclic star-autonomous categories; that is, unbraided star- autonomous categories in which the left and right duals of every object p are linked by coherent natural isomorphism. We settle coherence questions which have arisen concerning such cyclicity isomorphisms, and we show that such cyclic structures are the natural setting in which to consider enriched profunctors. Specifically, if V is a cyclic star-autonomous category, then the collection of V-enriched profunctors carries a canonical cyclic structure. In the case of braided star-autonomous categories, we discuss the correspon…
This paper presents a coherence theorem for star-autonomous categories exactly analogous to Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. The proof of this theorem is based on a categorial cut-elimination result, which is presented in some detail.
This paper gives a simple presentation of the free star-autonomous category over a category, based on Eilenberg-Kelly-MacLane graphs and Trimble rewiring, yielding a full coherence theorem: the commutativity of diagrams of canonical maps is decidable.
We study the question when a *-autonomous (Mix-)category has a representation as a $*$-autonomous category of a compact one. We prove that necessary and sufficient condition is that weak distributivity maps are monic (or, equivalently epic). For a Mix-category, this condition is, in turn, equivalent to the requirement that Mix-maps be monic (or epic). We call categories satisfying this property torsion-free. An important side result is that torsion-free categories have canonical partial traces.
Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing object.
We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. Indeed, any monoidal adjunction between autonomous categories gives rise to a Hopf monad. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke's criterium of semisimplicity, etc...) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again defined in a non-braided setting).
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in Prof, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced ∗-autonomous categories, proving a new equivalence result between the left ⊗-trace and the right ⅋-trace, and describing a new condition under which traced ͨ…
Traced monoidal categories as algebraic structures in $\mathbf{Prof}$
本研究提出迹伪幺半群的定义，给出柯西完备迹幺半范畴在$\u003cmath\u003e\mathbf{Prof}\u003c/math\u003e$中的新代数刻画，可借助图形演算推理迹，并将其用于研究迹$*$-自守范畴，得到迹的等价结果与范畴自守的新条件。
It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category i…
We study integrals of Hopf monoids in duoidal endohom categories of naturally Frobenius map monoidales in monoidal bicategories. We prove two Maschke type theorems, relating the separability of the underlying monoid and comonoid, respectively, to the existence of normalized integrals. It covers the examples provided by Hopf monoids in braided monoidal categories, weak Hopf algebras, Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal categories and Hopf categories.
Toy models have been used to separate important features of quantum computation from the rich background of the standard Hilbert space model. Category theory, on the other hand, is a general tool to separate components of mathematical structures, and analyze one layer at a time. It seems natural to combine the two approaches, and several authors have already pursued this idea. We explore *categorical comprehension construction* as a tool for adding features to toy models. We use it to comprehend quantum propositions and probabilities within the basic model of finite-dimensional Hilbert spaces.…
We develop an alternative approach to star-autonomous comonads via linearly distributive categories. It is shown that in the autonomous case the notions of star-autonomous comonad and Hopf comonad coincide.
A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. We define the term "quantum category". The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's star-autonomous categories is more suitable than autonomy (= compactness = rigidity). This leads to our definition of quantum groupoid intended as a "Hopf algebra with several objects".
We study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on the former. We also show that, conversely, if a Mix-category ${\bf K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of ${\bf K}$ into it, which induces the same mixed trace. Finally, we find a specif…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We define notions of functor and natural transformation for quantum categories.
We study the construction of premonoidal categories, where the pentagon relation fails, through representations of finite group algebras and their quantum doubles. Both finite group algebras and their quantum doubles have a finite number of irreducible representations. We show that in each case there are at least $2^{n-1}$ inequivalent premonoidal categories of representations, where $n$ is the number of irreducible representations. By construction, for the case of finite group algebras the categories are symmetric whereas for the quantum doubles the categories are braided.
Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasiz…
We describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a twisted quantum double of a finite group, this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of representation category of a twisted quantum double of a finite group. We also give a characterization of group-theoretical braided fusion …
There are various generalizations of bialgebras to their ''many object'' versions, such as quantum categories, bialgebroids and weak bialgebras. These can also be thought of as quantum analogues of small categories. In this paper we study modules over these structures, which are quantum analogues of profunctors (also called distributors) between small categories.
What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phung Ho Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstructed bialgebroid H has comodule category equivalent to the category of T-sheaves w.r.t. a monoidal Grothendieck topology on C. We also prove an existence theorem for fiber functors on small additive monoidal categories with bounded fusion and weak kernels. For certain autonomous categories a genera…
We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \otimes R.
Kornel Szlachányi recently used the term skew-monoidal category for a particular laxified version of monoidal category. He showed that bialgebroids $H$ with base ring $R$ could be characterized in terms of skew-monoidal structures on the category of one-sided $R$-modules for which the lax unit was $R$ itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal bicategory $\mathscr M$. These are skew-monoidal categories when $\mathscr M$ is $\mathrm{Cat}$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories in th…
We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $\mathfrak{u}_q(\mathfrak{g})$, where $q$ is a root of unity.
A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories were originally defined as monoidal comonads on endomorphism objects in a particular monoidal bicategory M. Then they were shown also to be skew monoidal structures (with an appropriate unit) on objects in M. Now we see in what kind of M quantum categories are merely monads.
Categories of quantum liquids II
本文发展可分高阶范畴理论及正交高阶范畴理论，推导出有隙量子液体的系统分类，并预测时空维度≥3中的新SPT序。
We formulate and study Howe-Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast}$-tensor categories. We say that a rigid $C^{\ast}$-tensor category $\mathcal{C}$ has the Howe-Moore property if every completely positive multiplier on $\mathcal{C}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe-Moore property. As an immediate consequence, we deduce the Howe-Moore property for Temperley-Lieb-Jones standard invariants with principal graph $A_…
This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int…
A quantum groups of type $A$ is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke symmetry.
Finite-dimensional quantum groups of type Super A and non-semisimple modular categories
本文研究了超A类型有限维量子群的构造及其非半单模范畴的分类，通过辫状德林费尔德双和非半单球形结构，提出了新的链接不变量并区分了某些结。
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…
We study the problem of convergence to the boundary in the setting of random walks on discrete quantum groups. Convergence to the boundary is established for random walks on $\hat{\textrm{SU}_q(2)}$. Furthermore, we will define the Martin boundary for random walks on C$^*$-tensor categories and give a formulation for convergence to the boundary for such random walks. These categorical definitions are shown to be compatible with the definitions in the quantum group case. This implies that convergence to the boundary for random walks on quantum groups is stable under monoidal equivalence.
We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general category-theoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set FRel.
This is a review article on modular categories, extending an invited talk given at the workshop "Categorical (co)algebraic methods in quantum informatics and linguistics", Oxford, October 29-31, 2010. To appear in C. Heunen, M. Sadrzadeh, E. Grefenstette (eds.): Compositional methods in quantum physics and linguistics, Oxford University Press, 2012.
Categories of quantum liquids I
本文基于凝聚完备化与拓扑 Wick 旋转建立可分离及酉高阶范畴理论，并统一刻画包含拓扑序、SPT/SET 序等的量子液体。
Involutive Markov categories and the quantum de Finetti theorem
研究对合马尔可夫范畴这一量子概念，以含预C* - 代数的范畴为主要示例，证明了关于最小和最大C* - 张量范数的量子德·菲内蒂定理并给出范畴描述。
In the present paper we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution structures. By utilizing category algebras and states on categories instead of simply considering categories, we can directly integrate relativity as a category theoretic structure and quantumness as a noncommutative probabilistic structure. Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQF…
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and internal groupoids. Over compact Hausdorff spaces, this lifting gives continuous symmetric encryption. Over a regular Mal'cev category, this correspondence gives a characterization of categories of completely positive maps, enabling the formulation of quantum features. These models are closer to Hilbert spaces than relations over sets in several respects: Heisenb…
We propose categories of $1$-dimensional and multi-dimensional quantum walks. In the categories, an object is a quantum walk, and a morphism is an intertwining operator between two quantum walks. The new framework enables us to discuss quantum walks in a unified way. The purposes of this paper are the following: (1) We reinterpret known results in our new framework. (2) We show several new theorems. For example, it is proved that every space-homogeneous time-periodic analytic quantum walk on $\mathbb{Z}^d$ has a limit distribution of velocity for every initial unit vector. Analyticity is a ver…
A collection of brilliant and original unfinished ideas by Wladyslaw Marcinek (1952-2003) in particle interactions, categorical approach to generalized statistics, qubits and quantum logic, entwined operators, cobordisms and noncommutative Fock space.
We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of finite-dimensional Hilbert spaces into the category of finite-dimensional C*-algebras and completely positive maps. In particular, the new category contains both quantum and classical channels, providing elegant abstract notions of preparation and measurement. We also consider nonstandard models, that can be used to investigate which notions from algebraic quantum information theory are operationally justifiable.
This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints…
We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses. We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process. We unif…
Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes sense to study applications of the theory of differential categories to categorical quantum foundations. In categorical quantum foundations, compact closed categories (and therefore traced symmetric monoidal categories) are one of the main objects of study, in particular the category of finite-dimensional Hilbert spaces FHilb. In this paper, we will explain w…
In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these m…
We are concerned with the center (=quantum double) of tensor categories and prove generalizations of several results proven previously for quantum doubles of Hopf algebras. We consider F-linear tensor categories C with simple unit and finitely many isomorphism classes of simple objects. We assume that C is either a *-category (i.e. there is a positive *-operation on the morphisms) or semisimple and spherical over an algebraically closed field F. In the latter case we assume that dim C=sum_i d(X_i)^2 is non-zero, where the summation runs over the isomorphism classes of simple objects. We prove:…
After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
This is the first paper in a series where we generalize the Categorical Quantum Mechanics program (due to Abramsky, Coecke, et al) to braided systems. In our view a uniform description of quantum information for braided systems has not yet emerged. The picture is complicated by a diversity of examples that lacks a unifying framework for proving theorems and discovering new protocols. We use category theory to construct a high-level language that abstracts the quantum mechanical properties of braided systems. We exploit this framework to propose an axiomatic description of braided quantum infor…
Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored $\ast$-operads and $\ast$-algebras. Central to the definition of colored $\ast$-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential $2$-adjunction whose right adjoint forms involutive functor categorie…
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