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Recently, there has been substantial interest in studying the dynamics of quantum theory beyond that of states, in particular, the dynamics of channels, measurements, and higher-order transformations. Ref. [Phys. Rev. X 8(1), 011047 (2018)]…
In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to…
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
We write down a series of basic laws for (strict) higher-order circuit diagrams. More precisely, we define higher-order circuit theories in terms of: (a) nesting, (b) temporal and spatial composition, and (c) equivalence between lower-order…
Recently there has been significant interest in using causal modelling techniques to understand the structure of physical theories. However, the notion of `causation' is limiting - insisting that a physical theory must involve causal…
Coecke and Heunen described completely positive maps in dagger monoidal categories and the {\sf CP}-infinity construction on these categories in order to construct a category of arbitrary dimensional quantum processes. This article…
Applied category theory often studies symmetric monoidal categories (SMCs) whose morphisms represent open systems. These structures naturally accommodate complex wiring patterns, leveraging (co)monoidal structures for splitting and merging…
Quantum operations are the most widely used tool in the theory of quantum information processing, representing elementary transformations of quantum states that are composed to form complex quantum circuits. The class of quantum…
Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories…
Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a…
Category theory provides a compact method of encoding mathematical structures in a uniform way, thereby enabling the use of general theorems on, for example, equivalence and universal constructions. In this article we develop the method of…
Constructor theory is a meta-theoretic approach that seeks to characterise concrete theories of physics in terms of the (im)possibility to implement certain abstract "tasks" by means of physical processes. Process theory, on the other hand,…
We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with…
We construct a monoidal category of open transition systems that generate material history as transitions unfold, which we call situated transition systems. The material history generated by a composite system is composed of the material…
We study preservation theorems for modal logics over finite structures with respect to three fundamental semantic relations: embeddings, injective homomorphisms, and homomorphisms. We focus on classes of pointed Kripke models that are…
We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$,…
It is well-known that combinatorial circuits are modeled mathematically by string diagrams in a monoidal category. Given a gate set $\Sigma$, the circuits over $\Sigma$ can be thought of as string diagrams in the free monoidal category…
We introduce a general notion of enrichment for homotopy-coherent algebraic structures described by Segal conditions, using the framework of "algebraic patterns" developed in our previous work. This recovers several known examples of…
It was recently suggested that causal structures are both dynamical, because of general relativity, and indefinite, due to quantum theory. The process matrix formalism furnishes a framework for quantum mechanics on indefinite causal…