Related papers: Universal Algebra and Effectful Computation

Substructural type systems, such as affine (and linear) type systems, are type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum…

A necessary and sufficient condition for an element of an algebra (in the sense of Universal Algebra) to be in the dominion of a subalgebra is given, in terms of transferable sets. This criterion is then used to formulate a more wieldy…

We give the definition of presentations of linear monoidal categories. Our main result is that given a presentation of a linear monoidal category, we can produce a presentation of the same category as a linear category. We apply this result…

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 study tensor categories that interpolate the representation categories of finite classical groups. There are (at least) two ways to approach these categories: via ultraproducts and via oligomorphic groups. Both have strengths and…

We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…

In this paper we construct a bicategory of (super) algebra bundles over a smooth manifold, where the 1-morphisms are bundles of bimodules. The main point is that naive definitions of bimodule bundles will not lead to a well-defined…

In previous work ("From signatures to monads in UniMath"), we described a category-theoretic construction of abstract syntax from a signature, mechanized in the UniMath library based on the Coq proof assistant. In the present work, we…

A new approach to the construction of general persistent polyhierarchical classifications is proposed. It is based on implicit description of category polyhierarchy by a generating polyhierarchy of classification criteria. Similarly to…

We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a…

This thesis develops the translation between category theory and computational linguistics as a foundation for natural language processing. The three chapters deal with syntax, semantics and pragmatics. First, string diagrams provide a…

Pseudoalgebras, introduced in [BDK], are multi-dimensional analogues of conformal algebras, which provide an axiomatic description of the singular part of the operator product expansion. Our main interest in this paper is the pseudoalgebra…

Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial…

The introduction of first-class type classes in the Coq system calls for re-examination of the basic interfaces used for mathematical formalization in type theory. We present a new set of type classes for mathematics and take full advantage…

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a *-structure, conjugate-linear on the…

This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…

We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories…

We define a notion on preadditive categories which plays a role similar to the notion of a Grothendieck pretopology on an unenriched category. Each such additive pretopology defines an additive Grothendieck topology and suffices to define…

Based on the monoid classifier, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and…

These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…