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A monoidal category where every object has a left and right adjoint is called a rigid category. String diagrams for rigid categories can be defined as non-progressive plane graphs, i.e. the edges can bend backward. In the context of categorical quantum mechanics, this is known as the snake equation.
This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects. commutes. A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal ...
Hybrid topology is also known as hybrid network. [19] Hybrid networks combine two or more topologies in such a way that the resulting network does not exhibit one of the standard topologies (e.g., bus, star, ring, etc.). For example, a tree network (or star-bus network) is a hybrid topology in which star networks are interconnected via bus ...
Ideas and tools from network science and engineering have been applied to the analysis of metabolic and genetic regulatory networks; the study of ecosystem stability and robustness; [11] clinical science; [12] the modeling and design of scalable communication networks such as the generation and visualization of complex wireless networks; [13 ...
The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds , with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak ...
A spatial network can be represented by a Voronoi diagram, which is a way of dividing space into a number of regions. The dual graph for a Voronoi diagram corresponds to the Delaunay triangulation for the same set of points. Voronoi tessellations are interesting for spatial networks in the sense that they provide a natural representation model ...
Such categories where the multiplicative monoidal structure is the categorical product and the additive monoidal structure is the coproduct are called distributive categories. Vect , the category of vector spaces over a field, with the direct sum as ⊕ {\displaystyle \oplus } and the tensor product as ⊗ {\displaystyle \otimes } .
The monoidal product of two diagrams is represented by placing one diagram above the other. Indeed, all ZX-diagrams are built freely from a set of generators via composition and monoidal product, modulo the equalities induced by the compact structure and the rules of the ZX-calculus given below.