Related papers: Extending Resource Monotones using Kan Extensions
A notion of morphism that is suitable for the sheaf-theoretic approach to contextuality is developed, resulting in a resource theory for contextuality. The key features involve using an underlying relation rather than a function between…
We review a recent development at the interface between discrete mathematics on one hand and probability theory and statistics on the other, specifically the use of Markov chains and their boundary theory in connection with the asymptotics…
This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the…
We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables having a particular…
In the first part of this article, we give an analysis of the free monad sequence in non-cocomplete categories, with the needed colimits explicitly parametrized. This enables us to state a more finely grained functoriality principle for…
Quantum complexity is emerging as a key property of many-body systems, including black holes, topological materials, and early quantum computers. A state's complexity quantifies the number of computational gates required to prepare the…
For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is particularly interesting to compare categories enriched in $\Q$ with…
We give an operational definition of information-theoretic resources within a given multipartite classical or quantum correlation. We present our causal model that serves as the source coding side of this correlation and introduce a novel…
The basic method of rewriting for words in a free monoid given a monoid presentation is extended to rewriting for paths in a free category given a `Kan extension presentation'. This is related to work of Carmody-Walters on the Todd-Coxeter…
The Grothendieck universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-sets or even the category of all locally U-small…
Via a family of monotone scalar functions, a preorder on a set is extended to its power set and then used to construct a hull operator and a corresponing complete lattice of sets. A function mappping into the preordered set is extended to a…
In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…
In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-Nets) constitute a useful framework with connections in both group theory and graph theory.…
The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms…
This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained…
Extensions of real numbers in more than two dimensions, in particular quaternions and octonions are finding applications in physics due to the fact that they naturally capture certain symmetries of physical systems. Here it is shown that…
The resource calculus is an extension of the lambda-calculus allowing to model resource consumption. It is intrinsically non-deterministic and has two general notions of reduction - one parallel, preserving all the possible results as a…
In addition to the important role of contextuality in foundations of quantum theory, this intrinsically quantum property has been identified as a potential resource for quantum advantage in different tasks. It is thus of fundamental…
Applications such as employees sharing office spaces over a workweek can be modeled as problems where agents are matched to resources over multiple rounds. Agents' requirements limit the set of compatible resources and the rounds in which…
We study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences. We explore the following problems: monotone complexity on product space, Lambalgen's theorem for correlated probability,…