Peter Selinger
We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces.
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Axiomatising the dagger category of complex Hilbert spaces
本文通过纯粹范畴条件公理化复希尔伯特空间的dagger范畴，提出简化方法以解决量子物理中的基础问题。
Hypergraph categories are symmetric monoidal categories where each object is equipped with a special commutative Frobenius algebra (SCFA). Dagger-hypergraph categories are the same, but with dagger-symmetric monoidal categories and dagger-SCFAs. In this paper, we show that finite matrices over a field K of characteristic 0 are complete for hypergraph categories, and that finite matrices where K has a non-trivial involution are complete for dagger-hypergraph categories.
Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories
针对若干看似关系范畴却不符合正则范畴关系理论的反常例子，本文在满正则独立范畴中建立平行关系理论，证明其与延迟 dagger 范畴的等价性，为解释这些例子提供新框架。
Characterizing (Co)Free Dagger Categories
本文通过内部dagger范畴结构，利用矩形带和之字形概念，分别给出了余自由与自由dagger范畴的充要刻画，并联系到伴随诱导的（余）单子（余）代数。
This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distr…
Limits in dagger categories
本研究提出dagger范畴的极限概念，证明其涵盖已知特例、酉同构下唯一等多项性质，明确其构造方式与交换性，丰富了dagger范畴的极限理论。
Unitary, Inner product, and Dagger categories
本文通过内积范畴重新定义了Dagger范畴，指出内积范畴等同于弱Dagger范畴，通过内积结构可将弱Dagger范畴严格化为Dagger范畴。
Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions
本研究揭示实数极限与范畴论极限的深层关联，基于已有全希尔伯特空间范畴刻画成果，用不含范数、实数等概念的范畴论结构完成有限维希尔伯特空间与线性压缩映射范畴的公理化刻画。
We define Frobenius-Eilenberg-Moore objects for a dagger Frobenius monad in an arbitrary dagger 2-category, and extend to the dagger context a well-known universal property of the formal theory of monads. We show that the free completion of a 2-category under Eilenberg-Moore objects extends to the dagger context, provided one is willing to work with such dagger Frobenius monads whose endofunctor part suitably commutes with their unit. Finally, we define dagger lax functors and dagger lax-limits of such functors, and show that Frobenius-Eilenberg-Moore objects are examples of such limits.
Monads on dagger categories
本文研究dagger范畴上的单子理论，证明满足Frobenius定律的dagger单子可分解为伴随且极值解仍具dagger结构，刻画了Frobenius定律的融贯性质及强单子与Frobenius幺半群的关系。
Models for dagger $(\infty,1)$-categories III: Univalent flagged dagger $(\infty,1)$-categories and unitarily complete dagger Segal anima
该文将 dagger 模型的 (∞,1)-范畴与万有带标记 dagger (∞,1)-范畴等同，构造伴随对证明其等价性，将万有性对应酉完备性，还等同了万有化与酉完备化。
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…
Models for dagger $(\infty,1)$-categories I: Dagger simplicial sets and unitary cores
本文为dagger (∞,1)-范畴构建单纯集模型，建立相关模型结构并证明伴随关系，引入酉核刻画等价关系，还与DCH反自伴随Joyal模型结构作比较。
We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative …
A dagger category is a category equipped with a functorial way of reversing morphisms, i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories with additional structure have been studied under different names e.g. in categorical quantum mechanics and algebraic field theory. In this thesis we study the dagger in its own right and show how basic category theory adapts to dagger categories. We develop a notion of a dagger limit that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isom…
Models for dagger $(\infty,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence
本文为dagger (∞,1)-范畴发展单纯空间模型，构造酉完备dagger Segal空间模型结构，并证明Joyal--Tierney等价的dagger类比，建立dagger Bergner、Joyal与Rezk模型间的Quillen等价链。
The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new category-theoretical tool, called the dagger-limit, which governs the way in which systems can be combined to form larger systems. These dagger-…
A coherence result for symmetric monoidal closed categories with biproducts is shown in this paper. It is explained how to prove, by using the same technique, coherence for compact closed categories with biproducts and for dagger compact closed categories with dagger biproducts.
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.…
O-categories generalize categories of domains to provide just the structure required to compute fixed points of locally continuous functors. Parametrized fixed points are of particular interest to denotational semantics and are often given by "dagger operations". We generalize existing techniques to define a functorial dagger operation on locally continuous functors between O-categories. We show that this dagger operation satisfies the Conway identities, a collection of identities used to axiomatize iteration theories. We study the behaviour of this dagger operation on natural transformations …
Dagger linear logic for categorical quantum mechanics
本文针对范畴量子力学中无限维过程推理的局限，提出以*-自伴和线性分配范畴推广匕首紧致闭范畴，并发展乘法匕首线性逻辑的范畴语义，最终定义混合酉范畴以扩展关键特征。
Non-Cartesian Guarded Recursion with Daggers
本文将对合范畴中的保护递归框架扩展到非笛卡尔环境，通过构造合适的范畴并研究其与对合范畴结构的交互，为高阶可逆编程语言添加保护递归特性。
Dagger Categories in Riemannian Geometry
本文在黎曼几何中研究dagger范畴，证明dagger编码度量与黎曼结构，分类微分算子范畴上的dagger，并揭示其与度量联络及格林公式的联系。
Dagger categories via anti-involutions and positivity
本文通过反自同态和正性概念，提出了一种新的 dagger 分类描述方法，证明反自同态分类与 dagger 分类在 2-范畴上是双等价的。
The humble $\dagger$ ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact. In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagge…
The spin-statistics theorem for topological quantum field theories
本文在Atiyah框架下通过引入费米子dagger紧致范畴，证明了拓扑量子场论的自旋-统计定理，并统一了自旋结构、超向量空间与幺正性。
Dagger $n$-categories
本文提出 dagger $(\infty,n)$-分类的概念，通过 bordism $(\infty,n)$-分类定义反射正拓扑量子场论，为量子场论基础提供更高范畴的框架。
Complete Positivity for Mixed Unitary Categories
本文把 dagger 幺半范畴的 CP-无穷构造推广到混合酉范畴，为任意维高阶量子编程语言提供语义，并证明该构造可由广义环境结构刻画。
We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative $\dagger$-Frobenius algebras, and we conclude $^\star\!\operatorname{Hilb}$ to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and po…
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…
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.
Convergence of martingales via enriched dagger categories
本文通过拓扑丰富类别理论，首次证明了鞅和逆向鞅在均值下的收敛性，并展示了其在概率分析中的应用。
The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we…
Complete W*-categories
该研究探讨W*-范畴，揭示完备W*-范畴的范畴化希尔伯特空间特性，构造典范范畴化内积，并证明完备W*-范畴间函子范畴的反线性等价关系。
Axioms for the category of sets and relations
本文为集合与关系的匕首范畴提供公理，其形式类似于近期关于Hilbert空间与有界算子匕首范畴的公理。
We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of dagger-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects …
Q-systems are unitary versions of Frobenius algebra objects which appeared in the theory of subfactors. In recent joint work with R. Hernández Palomares and C. Jones, the authors defined a notion of Q-system completion for C*/W* 2-categories, which is a unitary version of a higher idempotent completion in the spirit of Douglas--Reutter and Gaiotto--Johnson-Freyd. In this article, we prove that Q-system completion is a dagger 3-functor on the dagger 3-category of C*/W* 2-categories. We also prove that Q-system completion satisfies a universal property analogous to the universal property satisfi…
We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all finite (co)limits. An intermediate result that is important in its own right is that the scalars in such a category necessarily form an involutive field. In case of a Hilbert category, the embedding extends …
Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories
该研究提出准希尔伯特$*$-范畴作为阿贝尔范畴的希尔伯特空间类似物，基于对合反变自函子等结构推导其加性、拟阿贝尔等同调性质，并给出多个相关范畴实例。
We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-constru…
A diagrammatic calculus for categorical quantum protocols
本文提出一种基于一维配边和自由生成群的图形语言范畴，证明其具有双积的dagger紧致闭结构，并利用相干性结果，仅通过绘图即可验证量子隐形传态、纠缠交换和超密编码等量子协议的有效性。
We study the functor l^2 from the category of partial injections to the category of Hilbert spaces. The former category is finitely accessible, and its homsets are algebraic domains; the latter category has conditionally algebraic domains for homsets. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces.
Hilbert modules over $C^*$-categories
本文证明了$C^*$-范畴上Hilbert模的Eilenberg-Watts定理，通过imprimitivity双模张量积刻画了等价性，并应用于证明双范畴等价与Morita局部化。
Hilbert $*$-categories: Where limits in analysis and category theory meet
本文引入希尔伯特*范畴这一兼具代数与解析性质的范畴抽象，推导其两类解析完备性性质，依托等距映射宽子范畴的定向余极限构建两种新通用构造，为分析与范畴论极限的交汇提供了新的研究对象。
We investigate what would be a correct definition of categorical completeness for C*-categories and propose several variants of such a definition that make the category of Hilbert modules over a C*-algebra a free (co)completion. We extend results about generators and comparison theory known for W*-categories with direct sums and splitting of symmetric projections to our "complete C*-categories" and we give an abstract characterization of categories of Hilbert modules over a C*-algebra or a C*-category as "complete C*-category having enough absolutely compact morphisms (and a generator)". We th…
Hilbert series of representations of categories of $G$-sets
本文研究有限群 $G$ 的有限自由 $G$-集合范畴上反变函子的希尔伯特级数，证明其具有有理形式，且分母线性多项式的系数位于由 $G$ 的特征标表生成的域中。
Hilbert spaces over $C^*$-tensor categories, Fredholm modules and cyclic cohomology
本文在刚性 $C^*$-张量范畴上的广义 Hilbert 空间上构造 Fredholm 模，并利用 Connes 方法导出循环上同调中表现良好且仅依赖于同伦类的 Chern 特征。
C*-categories are essentially norm-closed *-categories of bounded linear operators between Hilbert spaces. The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gel'fand-Naimark representation theorem for Krein C*-algebras and Krein C*-categories.
** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete $C^{*}$-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of "non-degenerate" monoidal normal $*$-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand…
We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations''. The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. We introduce the notion of nuclear ideal to analyze these classes of morphisms. In compact closed categories, we see that all morphisms are nuclear, and in the category of Hilber…
A characterisation for the category of Hilbert spaces
针对现有Hilbert空间范畴刻画依赖monoidal结构而排除四元数的问题，本文提出一种无需该假设的替代刻画方法，成功统一了实数、复数与四元数Hilbert空间范畴。
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dime…
Model Categories and the Higher Riemann-Hilbert Correspondence
本研究通过右Bousfield局部化构造拓扑空间上dg预层范畴的新模型结构，证明光滑/复流形上德拉姆/Dolbeault代数的dg模与dg预层间的Quillen等价，还建立奇异上链代数dg反模与dg预层的同伦范畴等价，推进高阶黎曼-希尔伯特对应相关同伦理论研究。
We associate to each Temperley-Lieb-Jones C*-tensor category $\mathcal{T}\!\mathcal{L}\mathcal{J}(δ)$ with parameter $δ$ in the discrete range $\{2\cos(π/(k+2))\,:\,k=1,2,\ldots\}\cup\{2\}$ a certain C*-algebra $\mathcal{B}$ of compact operators. We use the unitary braiding on $\mathcal{T}\!\mathcal{L}\mathcal{J}(δ)$ to equip the category $\mathrm{Mod}_{\mathcal{B}}$ of (right) Hilbert $\mathcal{B}$-modules with the structure of a braided C*-tensor category. We show that $\mathcal{T}\!\mathcal{L}\mathcal{J}(δ)$ is equivalent, as a braided C*-tensor category, to the full subcategory $\mathrm{Mo…
We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in joint work with Crane as the basis for a representation theory of (Lie) 2-groups. Several important technical results are established along the way: First it is shown that all bounded invertible additive functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert s…
Unitary TQFTs, unitary disk-like $n$-categories, and higher Hilbert spaces
本文提出用完备有限酉圆盘状$n$-范畴定义有限$(n+1)$-希尔伯特空间，验证了$n=1,2$情形，并由此分类了低维酉TQFT。
Control theory uses "signal-flow diagrams" to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the symmetric monoidal category FinVect_k of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVect_k in terms of the generators used in signal …
The many faces of higher Hilbert spaces
本文通过G- dagger范畴统一了有限维算子代数作为C*, W*, H*代数时的模范畴与对应2-范畴差异，引入G- Hermitian 2-向量空间并定义正性条件，为高维希尔伯特空间提供归纳定义框架。
We investigate notions of ambiguity and partial information in categorical distributional models of natural language. Probabilistic ambiguity has previously been studied using Selinger's CPM construction. This construction works well for models built upon vector spaces, as has been shown in quantum computational applications. Unfortunately, it doesn't seem to provide a satisfactory method for introducing mixing in other compact closed categories such as the category of sets and binary relations. We therefore lack a uniform strategy for extending a category to model imprecise linguistic informa…
Normal categories of normed algebra of finite rank bounded operators
本文研究有限秩有界算子赋范代数的主理想正规范畴，证明左右正规范畴同构，并刻画其正规锥半群及有界正规锥构成的赋范代数与原代数同构。
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