In mathematics, a monoidal category (or tensor category) is a bicategory with one object. More explicitly, a monoidal category is a category $\backslash mathbb\; C$ equipped with a binary functor

$\backslash otimes:\; \backslash mathbb\; C\backslash times\backslash mathbb\; C\backslash to\backslash mathbb\; C$

called tensor, and a unit object $I$.

A monoidal category must be equipped with three natural isomorphisms expressing the fact that the tensor operation should

• be associative: there is a natural isomorphism $\backslash alpha$, called associativity, with components

$\backslash alpha\_\{A,B,C\}:\; (A\backslash otimes\; B)\backslash otimes\; C\; \backslash to\; A\backslash otimes(B\backslash otimes\; C)$,

• have $I$ as left and right identity: there are two natural isomorphisms $\backslash lambda$ and $\backslash rho$, respectively called left and right identity, with components

$\backslash lambda\_A:\; I\backslash otimes\; A\backslash to\; A$

and

$\backslash rho\_A:\; A\backslash otimes\; I\backslash to\; A$.

These natural transformations are subject to certain coherence conditions. All the necessary conditions are implied by the following two: for all $A$, $B$, $C$ and $D$ in $\backslash mathbb\; C$, the diagrams and must commute.

It follows from these two conditions that any such diagram (i.e. a diagram whose morphisms are built using $\backslash alpha$, $\backslash lambda$, $\backslash rho$, 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.

Strict monoidal categories

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A monoidal category is said to be a strict monoidal category when the natural isomorphisms $\backslash alpha$, $\backslash lambda$ and $\backslash rho$ are identities.

For every category $\backslash mathbb\; C$, the free strict monoidal category $\backslash Sigma(\backslash mathbb\; C)$ can be constructed as follows:

• its objects are lists (finite sequences) $A\_1,\backslash ldots,\; A\_n$ of objects of $\backslash mathbb\; C$,
• there are arrows between two objects $A\_1,\backslash ldots,\; A\_n$ and $B\_1,\backslash ldots,\; B\_m$ if and only if $n=m$ and in this case the arrows are lists (finite sequences) of arrows $f\_1:A\_1\backslash to\; B\_1,\; \backslash ldots,\; f\_n:A\_n\backslash to\; B\_n$ of $\backslash mathbb\; C$,
• the tensor product of two objects $A\_1,\backslash ldots,\; A\_n$ and $B\_1,\backslash ldots,\; B\_m$ is the concatenation $A\_1,\backslash ldots,\; A\_n,\; B\_1,\backslash ldots,\; B\_m$ of the two lists and similarly the tensor product of two morphisms is given by the concatenation of lists.

This operation $\backslash Sigma$ which to a category $\backslash mathbb\; C$ associates $\backslash Sigma(\backslash mathbb\; C)$ can be extended into a strict 2-monad on $\backslash textbf\{Cat\}$.

Examples

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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.

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$R$-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 ⊗ 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 $\backslash nabla:A\backslash otimes\; A\backslash rightarrow\; A$ and $\backslash eta:\; R\; \backslash rightarrow\; A$ satisfying	A monoid is an object M together with morphisms $\backslash circ:\; M\; \backslash times\; M\; \backslash rightarrow\; M$ and $1:\; \backslash \{*\backslash \}\; \backslash rightarrow\; M$ satisfying
and	and
A coalgebra is an object C with morphisms $\backslash Delta:\; C\; \backslash rightarrow\; C\; \backslash otimes\; C$ and $\backslash epsilon:C\backslash rightarrow\; R$ satisfying	Any object of Set, S has two unique morphisms $\backslash Delta:\; S\; \backslash rightarrow\; S\; \backslash times\; S$ and $\backslash epsilon:\; S\; \backslash rightarrow\; \backslash \{*\backslash \}$ satisfying
and	and
	In particular, ε is unique because $\backslash \{*\backslash \}$ is a terminal object.

See also

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• 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).

References

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• 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 (1997), Categories for the Working Mathematician (2nd ed.). New York: Springer-Verlag.

Monoidal categories

Monoidale Kategorie