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 such that
 * a bifunctor $$\otimes\colon\mathcal M\times\mathcal M\to\mathcal M$$, and
 * an object $$I$$ of $$\mathcal M$$ called the unit object
 * $$(-\otimes -)\otimes - = -\otimes(-\otimes -)$$ as trifunctors $$\mathcal M\times\mathcal M\times\mathcal M\to\mathcal M$$, and
 * $$I\otimes - = 1_{\mathcal M} = -\otimes I$$ as functors $$\mathcal M\to\mathcal M$$.

Alternate formulations
A strict monoidal category can be defined as
 * 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

Examples

 * 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 $$\mathcal C$$.

Free strict monoidal category
For every category C, the free strict monoidal category Σ(C) can be constructed as follows: This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat.
 * 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: A1 → B1, ..., fn: An → Bn 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.