Related papers: Dagger Categories of Tame Relations

We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give…

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…

A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is…

We develop the idea of a supersymmetric monoidal supercategory, following ideas of Kapranov. Roughly, this is a monoidal category in which the objects and morphisms are ${\bf Z}/2$-graded, equipped with isomorphisms $X \otimes Y \to Y…

The aim of this article is to study certain categorical-algebraic frameworks for basic homological algebra, introduced in arXiv:2404.15896, with the aim of better understanding the differences between them. We focus on homological…

We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we…

An orthoset is a non-empty set together with a symmetric and irreflexive binary relation $\perp$, called the orthogonality relation. An orthoset with 0 is an orthoset augmented with an additional element 0, called falsity, which is…

Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can…

A symmetric monoidal category is a category equipped with an associative and commutative (binary) product and an object which is the unit for the product. In fact, those properties only hold up to natural isomorphisms which satisfy some…

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the…

This paper contains two new results: 1. We amend the notion of abstract basis in a dagger symmetric monoidal category, as well as its corresponding graphical representation, in order to accommodate non-self-dual dagger compact structures;…

One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization…

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…

Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular,…

We extend the usual process-theoretic view on locality and causality in subsystems (based on the tensor product case) to general quantum systems (i.e.\ possibly non-factor, finite-dimensional von Neumann algebras). To do so, we introduce a…

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every…

Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable $(\infty,1)$-categories, while the…

Main contribution of this paper is an investigation of expressive power of the database category DB. An object in this category is a database-instance (set of n-ary relations). Morphisms are not functions but have complex tree structures…

In this paper, we introduce the notion of a von Neumann category, as a generalization and categorification of von Neumann algebra. A von Neumann category is a premonoidal category with compatible dagger structure which embeds as a double…

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive…