Related papers: Q-system Completion of 2-functors
We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is considered as a…
The functionals on an ordered semigroup S in the category Cu--a category to which the Cuntz semigroup of a C*-algebra naturally belongs--are investigated. After appending a new axiom to the category Cu, it is shown that the "realification"…
Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*-algebra A equipped with an orthogonal filtration as the universal object in the category of compact…
We introduce a category of inverse semigroup actions and a category of \'etale groupoids. We show that there are three functors which send inverse semigroups to their spectral actions, inverse semigroup actions to their transformation…
We prove that a haploid associative algebra in a $C^*$-tensor category $\mathcal{C}$ is equivalent to a Q-system (a special $C^*$-Frobenius algebra) in $\mathcal{C}$ if and only if it is rigid. This allows us to prove the unitarity of all…
This work is largely focused on extending D. Higgs' $\Omega$-sets to the context of quantales, following the broad program of U. H\"ohle, we explore the rich category of $\mathscr Q$-sets for strong, integral and commutative quantales, or…
We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of 1-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted…
Let $G$ be a finite group and, for a given complex character $\chi$ of $G$, let ${\mathbb{Q}}(\chi)$ denote the field extension of ${\mathbb{Q}}$ obtained by adjoining all the values $\chi(g)$, for $g\in G$. The group $G$ is called…
We show that the double category $\mathbb{C}\mathbf{at}^\#$ of comonoids in the category of polynomial functors (previously shown by Ahman-Uustalu and Garner to be equivalent to the double category of categories, cofunctors, and…
Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $\omega$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the…
The representations of a quiver Q over a field k have been studied for a long time. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when…
The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(\infty). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which…
Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding…
A functor of sets $\mathbb X$ over the category of $K$-commutative algebras is said to be an affine functor if its functor of functions, $\mathbb A_{\mathbb X}$, is reflexive and $\mathbb X=\Spec \mathbb A_{\mathbb X}$. We prove that affine…
Observables in a quantum system, represented by a Hilbert space, are given by the orthogonal bases of the aforementioned Hilbert space. Categorical Quantum Mechanics provides further abstraction of such observables, allowing for a…
We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
We define united K-theory for real C*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural…
We review the notions of a multiplier category and the $W^{*}$-envelope of a $C^{*}$-category. We then consider the notion of an orthogonal sum of a (possibly infinite) family of objects in a $C^{*}$-category. Furthermore, we construct…
We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion. Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their…