Jump to content

User:Fropuff/Drafts/Strict monoidal category

From Wikipedia, the free encyclopedia
Official page: Strict monoidal category

In mathematics, especially in category theory, a strict monoidal category is a category with a unital and associative bifunctor .

Formal definition

[edit]

A 'strict monoidal category is a category together with

  • a bifunctor , and
  • an object of called the unit object

such that

  • as trifunctors , and
  • as functors .

Alternate formulations

[edit]

A strict monoidal category can be defined as

1 Set
Monoid Category
Strict monoidal category 2-category
n-monoid n-category

Examples

[edit]
  • A monoid is essentially the same thing as discrete strict monoidal category.
  • Every bounded semilattice , considered as a thin category, is strict monoidal with serving as the product and 1 as the unit.
  • A strict monoidal category with a single object is essentially a commutative monoid. This follows from the Eckmann–Hilton argument. A lax monoidal category with a single object is necessarily strict.
  • The (augmented) simplex category is strict monoidal with addition of ordinals serving as the monoidal product.
  • Given any preordered set , the set of all endomorphisms of (i.e. monotone functions ) forms a strict monoidal category with composition serving as the product and the identity map as the unit.
  • The cateogry of endofunctors, , on a given category form a strict monoidal category with composition of endofunctors serving as the product and the identity functor serving as the unit. This reduces to the previous case when is thin.
  • Given any (strict) 2-category , the endomorphisms of any object in form a strict monoidal category. Actually, every example is of this form (see below).
  • Given any category we can form the free strict monoidal category on .

Free strict monoidal category

[edit]

For every category C, the free strict monoidal category Σ(C) can be constructed as follows:

  • its objects are lists (finite sequences) A1, ..., An of objects of C;
  • there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m = n, and then the arrows are lists (finite sequences) of arrows f1: A1B1, ..., fn: AnBn of C;
  • the tensor product of two objects A1, ..., An and B1, ..., Bm is the concatenation A1, ..., An, B1, ..., Bm of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists. The identity object is the empty list.

This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat.