User:Fropuff/Drafts/Strict monoidal category

Official page: Strict monoidal category

In mathematics, especially in category theory, a strict monoidal category is a category ${\mathcal {M}}$ with a unital and associative bifunctor $\otimes \colon {\mathcal {M}}\times {\mathcal {M}}\to {\mathcal {M}}$ .

Formal definition

A 'strict monoidal category is a category ${\mathcal {M}}$ together with

a bifunctor $\otimes \colon {\mathcal {M}}\times {\mathcal {M}}\to {\mathcal {M}}$ , and
an object $I$ of ${\mathcal {M}}$ called the unit object

such that

$(-\otimes -)\otimes -=-\otimes (-\otimes -)$ as trifunctors ${\mathcal {M}}\times {\mathcal {M}}\times {\mathcal {M}}\to {\mathcal {M}}$ , and
$I\otimes -=1_{\mathcal {M}}=-\otimes I$ as functors ${\mathcal {M}}\to {\mathcal {M}}$ .

a monoid object in Cat
- a category with an associative, unital bifunctor $\otimes$
an internal category in Mon
- a 2-monoid (a monoid with a monoid of arrows between elements satisfying certain axioms)
a (strict) 2-category with a single object

A monoid is essentially the same thing as discrete strict monoidal category.
Every bounded semilattice $\langle S,\land ,1\rangle$ , considered as a thin category, is strict monoidal with $\land$ 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 $\Delta$ is strict monoidal with addition of ordinals serving as the monoidal product.
Given any preordered set $P$ , the set $\operatorname {End} (P)$ of all endomorphisms of $P$ (i.e. monotone functions $P\to P$ ) forms a strict monoidal category with composition serving as the product and the identity map $\operatorname {id} _{P}$ as the unit.
The cateogry of endofunctors, $\operatorname {End} ({\mathcal {C}})$ , on a given category ${\mathcal {C}}$ form a strict monoidal category with composition of endofunctors serving as the product and the identity functor $1_{\mathcal {C}}$ serving as the unit. This reduces to the previous case when ${\mathcal {C}}$ is thin.
Given any (strict) 2-category ${\mathcal {E}}$ , the endomorphisms of any object in $C$ form a strict monoidal category. Actually, every example is of this form (see below).
Given any category ${\mathcal {C}}$ we can form the free strict monoidal category on ${\mathcal {C}}$ .

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

its objects are lists (finite sequences) A₁, ..., A_n of objects of C;
there are arrows between two objects A₁, ..., A_m and B₁, ..., B_n only if m = n, and then the arrows are lists (finite sequences) of arrows f₁: A₁ → B₁, ..., f_n: A_n → B_n of C;
the tensor product of two objects A₁, ..., A_n and B₁, ..., B_m is the concatenation A₁, ..., A_n, B₁, ..., B_m 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.