Related papers: Composable Uncertainty in Symmetric Monoidal Categ…
Applied category theory provides powerful mathematical tools for modelling processes and their composition. Symmetric monoidal categories, which involve series and parallel composition, are particularly well-suited for describing the…
Matching 't Hooft anomalies is a powerful tool for constraining the low-energy dynamics of quantum systems and their allowed renormalization group (RG) flows. For non-invertible (or categorical) symmetries, however, a key challenge has been…
Working in the framework of $(T, V)$-categories, for a symmetric monoidal closed category $V$ and a (not necessarily cartesian) monad $T$, we present a common account to the study of ordered compact Hausdorff spaces and stably compact…
A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories…
This paper develops a systematic framework for integrating local categories that model logical connectives using higher category theory. By extending these local categories into a unified two-category enriched with natural isomorphisms, the…
In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that…
In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits…
We introduce some algebraic structures such as singularity, commutators and central extension in modified categories of interest. Additionally, we introduce the cat$^{1}$-objects with their connection to crossed modules in these categories…
Field induced assembly of reconfigurable structures with complex hierarchical configurations has recently become an area of intense research with the promise for exciting applications in programmable self-assembly and nano/microstructure…
The purpose of this note is to show that, if $\mathcal{V}$ is a closed monoidal category, the following three notions are equivalent. (1) Category with $\mathcal{V}$-structure and cylinder. (2) Tensored $\mathcal{V}$-category. (3)…
Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by…
We address the (pointed) homotopy of crossed module morphisms in modified categories of interest; which generalizes the groups and various algebraic structures. We prove that, the homotopy relation gives rise to an equivalence relation;…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as…
This paper provides a mathematical framework for client-server communication that results in a modular and type-safe architecture. It is informed and motivated by the software engineering practice of developing server backends with a…
Feature toggles and configuration options are modern programmatic techniques to easily include or exclude functionality in a software product. The research contributions to these two techniques have most often been focused on either one of…
We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skew-commutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for…
We sketch a procedure to capture general non-invertible symmetries of a d-dimensional quantum field theory in the data of a higher-category, which captures the local properties of topological defects associated to the symmetries. We also…
Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them.…
Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category…