Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInduced monoidal structure from the functor
60
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $mathcal{B}$ be a subcategory of a given category $mathcal{D}$. Let $mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $mathcal{B}$ to $mathcal{D}$ such that $mathcal{B}$ becomes a sub monoidal category of monoidal category $mathcal{D}$. Examples are discussed, and in particular, in an example of loop space, we elaborated all results discussed in this article.
rate research
Read More
The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation. Spacetime structure on the base space induces a closure operator on the idempotent subunits. Restriction is then interpreted as spacetime propagation. This lets us study relativistic quantum information theory using methods entirely internal to monoidal categories. As a proof of concept, we show that quantum teleportation is only successfully supported on the intersection of Alice and Bobs causal future.
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and we use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, seen as 3-dimensional word problems in a track category, including the case of braided monoidal categories.
We introduce DisCoPy, an open source toolbox for computing with monoidal categories. The library provides an intuitive syntax for defining string diagrams and monoidal functors. Its modularity allows the efficient implementation of computational experiments in the various applications of category theory where diagrams have become a lingua franca. As an example, we used DisCoPy to perform natural language processing on quantum hardware for the first time.
This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.
For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (apparently) not part of the standard introductory treatments of this subject.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
