Related papers: Supermaps on generalised theories
We establish and advocate for a novel branch of category theory, centered around strong dinatural transformations (herein known as "paranatural transformations"). Paranatural transformations generalize natural transformations to…
We propose a new line of attack to create a finite quantum theory which includes general relativity and (perhaps) the standard model in its low energy limit. The theory would emerge from the categorical approach. A structure is observed on…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…
We argue that notions in quantum theory should have universal properties in the sense of category theory. We consider the completely positive trace preserving (CPTP) maps, the basic notion of quantum channel. Physically, quantum channels…
In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of…
We observe that the existence of sequential and parallel composition supermaps in higher order theories of transformations can be formalised using enriched category theory. Encouraged by relevant examples such as unitary supermaps and…
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can…
Parameterization extends higher-order processes with the capability of abstraction and application (like those in lambda-calculus). This extension is strict, i.e., higher-order processes equipped with parameterization is computationally…
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
We reconstruct finite-dimensional quantum theory with superselection rules, which can describe hybrid quantum-classical systems, from four purely operational postulates: symmetric sharpness, complete mixing, filtering, and local equality.…
A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties…
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such…
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual…
Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…
We consider anti-unification for simply typed lambda terms in associative, commutative, and associative-commutative theories and develop a sound and complete algorithm which takes two lambda terms and computes their generalizations in the…
I used to believe that my conventions for drawing diagrams for categorical statements could be written down in one page or less, and that the only tricky part was the technique for reconstructing objects "from their names"... but then I…
We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an…
We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as a potential yoga for dealing with processes and processes of…
We prove a general congruence result for bisimilarity in higher-order languages, which generalises previous work to languages specified by a labelled transition system in which programs may occur as labels, and which may rely on operations…