Optics are a data representation for compositional data access, with lenses as a popular special case. Hedges has presented a diagrammatic calculus for lenses, but in a way that does not generalize to other classes of optic. We present a calculus that works for all optics, not just lenses; this is done by embedding optics into their presheaf category, which naturally features string diagrams. We apply our calculus to the common case of lenses, extend it to effectful lenses, and explore how the laws of optics manifest in this setting.
A popular graphical calculus for monoidal categories makes computations tactile and intuitive. Complicated diagram chases can be expressed in a few pictures and discovered by playing with a shoelace. Joyal and Street's proof of the soundness of this calculus says that any deformation of a diagram, any bending of the strings, describes the same morphism. In this paper, we extend the graphical calculus to double categories and proarrow equipments in order to bring the string diagrammatic method to formal category theory. Our main theorem proves this calculus sound with the help of Dawson and Par…
String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less attention has been given to their algorithmic properties, and efficient implementations of diagrammatic reasoning are almost an unexplored subject. This work intends to be a contribution in such a direction. We introduce a data structure representing string diagrams in terms of adjacency matrices. This encoding has the key advantage of providing simple and effi…
Bimonoidal categories (also known as rig categories) are categories with two monoidal structures, one of which distributes over the other. We formally define sheet diagrams, a graphical calculus for bimonoidal categories that was informally introduced by Staton. Sheet diagrams are string diagrams drawn on a branching surface, which is itself an extruded string diagram. Our main result is a soundness and completeness theorem of the usual form for graphical calculi: we show that sheet diagrams form the free bimonoidal category on a signature.
We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob/O in the sense that there is an equivalence between (Cob/O)-algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms …
A Critical Pair Enumeration Algorithm for String Diagram Rewriting
Johns Hopkins University（约翰霍普金斯大学）；University of Washington（华盛顿大学）；University of Technology of Compiègne（贡比涅技术大学）；Universidad Nacional de Córdoba（科尔多瓦国立大学）；Tallinn University of Technology（塔林理工大学）；University of São Paulo（圣保罗大学）；National Institute of Informatics（日本国立情报学研究所）；Ochanomizu University（御茶水女子大学）
These are lectures notes for a mini-course given at the conference Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras, and Categorification in June 2018. The goal is to introduce the reader to string diagram techniques for monoidal categories, with an emphasis on their role in categorification.
We define a monad $T_n^{\operatorname{D^s}}$ whose operations are encoded by simple string diagrams and we define $n$-sesquicategories as algebras over this monad. This monad encodes the compositional structure of $n$-dimensional string diagrams. We give a generators and relations description of $T_n^{\operatorname{D^s}}$ , which allows us to describe $n$-sesquicategories as $n$-globular sets equipped with associative and unital composition and whiskering operations. One can also see them as strict $n$-categories without interchange laws. Finally we give an inductive characterization of $n$-se…
String diagrams are a powerful and intuitive graphical syntax, originated in the study of symmetric monoidal categories. In the last few years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In many such proposals, the transformations of the described systems are modelled as rewrite rules of diagrams. These developments demand a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of stri…
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thoughts as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory…
Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different d…
Radboud University（拉德堡德大学）；University College London（伦敦大学学院）
Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effec…
Linear/non-linear (LNL) models, as described by Benton, soundly model a LNL term calculus and LNL logic closely related to intuitionistic linear logic. Every such model induces a canonical enrichment that we show soundly models a LNL lambda calculus for string diagrams, introduced by Rios and Selinger (with primary application in quantum computing). Our abstract treatment of this language leads to simpler concrete models compared to those presented so far. We also extend the language with general recursion and prove soundness. Finally, we present an adequacy result for the diagram-free fragmen…
Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy
Quantinuum（昆蒂努姆）；Association for Mathematical Consciousness Science（数学意识科学协会）；Ludwig Maximilian University of Munich（慕尼黑路德维希-马克西米利安大学）；University of Bamberg（班贝格大学）；Topos Institute（托波斯研究所）；VERSES Research（VERSES研究院）
