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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.
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 ,
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 ...
Bord n is a monoidal category under the operation which maps two bordisms to the bordism made from their disjoint union. A TQFT on n -dimensional manifolds is then a functor from hBord n to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product.
The technical advantage of the category of symmetric spectra is that it has a closed symmetric monoidal structure (with respect to smash product). It is also a simplicial model category . A symmetric ring spectrum is a monoid in S p Σ {\displaystyle {\mathcal {S}}p^{\Sigma }} ; if the monoid is commutative, it's a commutative ring spectrum .
A set S equipped with a binary operation S × S → S, which we will denote •, is a monoid if it satisfies the following two axioms: . Associativity For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds.
Andrew Casson introduces the Casson invariant for homology 3-spheres, bringing the whole new set of ideas into the 3-dimensional topology, and relating the geometry of 3-manifolds with the geometry of representation spaces of the fundamental group of a 2-manifold. This leads to a direct connection with mathematical physics.