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In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's binary operation is continuous. Every topological group is a topological monoid.
When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.
A monoidal category is a category equipped with a monoidal structure. A monoidal structure consists of the following: A monoidal structure consists of the following: a bifunctor ⊗ : C × C → C {\displaystyle \otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C} } called the monoidal product , [ 2 ] or tensor product ,
Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates , or signal types may differ between two different networks, yet their logical ...
A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale. A monoid object in (Ab, ⊗ Z, Z), the category of abelian groups, is a ring. For a commutative ring R, a monoid object in (R-Mod, ⊗ R, R), the category of modules over R, is a R-algebra.
Quantum categories and quantum groupoids: A quantum category over a braided monoidal category V is an object R with an opmorphism h: R op ⊗ R → A into a pseudomonoid A such that h * is strong monoidal (preserves tensor product and unit up to coherent natural isomorphisms) and all R, h and A lie in the autonomous monoidal bicategory Comod(V ...
Categories include sets, groups and topologies. Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept ...
The category of small categories is a closed monoidal category in exactly two ways: with the usual categorical product and with the funny tensor product. [6] Given two categories and , let be the category with functors,: as objects and unnatural transformations: as arrows, i.e. families of morphisms {: ()} which do not necessarily satisfy the condition for a natural transformation.