Monoidal category

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In mathematics, a monoidal category (or tensor category) is a category $\mathbf C$ equipped with

a binary functor $\otimes: \mathbf C\times\mathbf C\to\mathbf C$ called the tensor product,
an object $I$ called the unit object,
three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation
- is associative: there is a natural isomorphism $α$ , called associativity, with components $\alpha_{A,B,C}: (A\otimes B)\otimes C \to A\otimes(B\otimes C)$ ,
- has $I$ as left and right identity: there are two natural isomorphisms $λ$ and $ρ$ , respectively called left and right identity, with components $\lambda_A: I\otimes A\to A$ and $\rho_A: A\otimes I\to A$ .

The coherence conditions for these natural transformations are the following two: for all $A$ , $B$ , $C$ and $D$ in $\mathbf C$ , the diagrams

and

both commute.

It follows from these two conditions that any such diagram (i.e. a diagram whose morphisms are built using $α$ , $λ$ , $ρ$ , identities and tensor product) commutes: this is Mac Lane's "coherence theorem". This is related to the fact that every monoidal category is monoidally equivalent to a strict (see below) monoidal category.

Monoidal categories are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter.

1 Strict monoidal categories
2 Examples
3 See also
4 References

[edit] Strict monoidal categories

A monoidal category is said to be a strict monoidal category when the natural isomorphisms α, λ and ρ are identities.

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 if and 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.

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

[edit] Examples

Any category with standard categorical products and a terminal object is a monoidal category, with the categorical product as tensor product and the terminal object as identity. Also, any category with coproducts and an initial object is a monoidal category - with the coproduct as tensor product and the initial object as identity. (In both these cases, the structure is actually symmetric monoidal.) However, in many monoidal categories (such as $R$ -Mod, given below) the tensor product is neither a categorical product nor a coproduct.

Examples of monoidal categories, illustrating the parallelism between the category of vector spaces over a field and the category of sets, are given below.

$R$ -Mod	Set
Given a field or commutative ring $R$ , the category $R$ -Mod of $R$ -modules (in the case of a field, vector spaces) is a symmetric monoidal category with product $\otimes$ and identity $R$ .	The category Set is a symmetric monoidal category with product × and identity {*}.
A unital associative algebra is an object of $R$ -Mod together with morphisms $\nabla:A\otimes A\rightarrow A$ and $\eta: R \rightarrow A$ satisfying	A monoid is an object M together with morphisms $\circ: M \times M \rightarrow M$ and $1: \{*\} \rightarrow M$ satisfying

and	and
.	.
A coalgebra is an object C with morphisms $\Delta: C \rightarrow C \otimes C$ and $\epsilon:C\rightarrow R$ satisfying	Any object of Set, S has two unique morphisms $\Delta: S \rightarrow S \times S$ and $\epsilon: S \rightarrow \{*\}$ satisfying

and	and
.	.
	In particular, ε is unique because ${ * }$ is a terminal object.

[edit] See also

Many monoidal categories have additional structure such as braiding, symmetry or closure: the references describe this in detail.
Monoidal functors are the functors between monoidal categories 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 monoidal category, which generalizes the ordinary notion of monoid. In particular, a strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product).
A monoidal category can also be seen as the category B(□, □) of a bicategory B with only one object, denoted □.
Bounded-above meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element.

[edit] References

Mac Lane, Saunders (1963). "Natural Associativity and Commutativity". Rice University Studies 49, 28–46.
Kelly, G. Max (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc." Journal of Algebra 1, 397–402
Joyal, André; Street, Ross (1993). "Braided Tensor Categories". Advances in Mathematics 102, 20–78.
Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.). New York: Springer-Verlag.