A symmetric monoidal category is a monoidal category which is commutative up to a natural isomorphism. Formally, it is a braidedmonoidal category whose braiding satisfies In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Braided monoidal category is a mathematical concept in terms of category theory and is, as its name suggests, a monoidal category with braiding. ... In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). ...
In mathematics, a monoidalcategory (or tensor category) is a bicategory with one object.
Monoidal functors are the functors between monoidalcategories which preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.
There is a general notion of monoid object in a monoidalcategory, which generalizes the ordinary notion of monoid.
In category theory, a PRO is a strict monoidalcategory whose objects are the natural integers and whose tensor product is given on objects by the addition on integers.
An algebra of a PRO P in a category C is a strict monoidal functor from P to C.
More precisely, what we mean here by "the algebras of Δ in C are the monoid objects in C" for example is that the category of algebras of P in C is equivalent to the category of monoids in C.
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