In mathematics, a **monoidal category** (or **tensor category**) is a category equipped with a bifunctor

- Formal definition
- Examples
- Monoidal preorders
- Properties and associated notions
- Free strict monoidal category
- Specializations
- See also
- References

that is associative up to a natural isomorphism, and an object *I* that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute.

The ordinary tensor product makes vector spaces, abelian groups, *R*-modules, or *R*-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product.

A rather different application, of which monoidal categories can be considered an abstraction, is that of a system of data types closed under a type constructor that takes two types and builds an aggregate type; the types are the objects and is the aggregate constructor. The associativity up to isomorphism is then a way of expressing that different ways of aggregating the same data—such as and —store the same information even though the aggregate values need not be the same. Identity objects are analogous to algebraic operations addition (type sum) and multiplication (type product). For type product - identity object is the unit , it trivially fully inhabits its type, so there is only one inhabitant of the type, and that is why a product with it is always isomorphic to the other operand. For type sum, the identity object is the void type, which stores no information and its inhabitants impossible to address. The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory.^{ [1] }

In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category.

Monoidal categories have numerous applications outside of category theory proper. They 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. Braided monoidal categories have applications in quantum information, quantum field theory, and string theory.

A **monoidal category** is a category equipped with a monoidal structure. A monoidal structure consists of the following:

- a bifunctor called the
*tensor product*or*monoidal product*, - an object called the
*unit object*or*identity object*, - three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation
- is associative: there is a natural (in each of three arguments , , ) isomorphism , called
*associator*, with components , - has as left and right identity: there are two natural isomorphisms and , respectively called
*left*and*right unitor*, with components and .

- is associative: there is a natural (in each of three arguments , , ) isomorphism , called

Note that a good way to remember how and act is by alliteration; *Lambda*, , cancels the identity on the *left*, while *Rho*, , cancels the identity on the *right*.

The coherence conditions for these natural transformations are:

- for all , , and in , the pentagon diagram

- commutes;

- for all and in , the triangle diagram

- commutes.

A **strict monoidal category** is one for which the natural isomorphisms *α*, *λ* and *ρ* are identities. Every monoidal category is monoidally equivalent to a strict monoidal category.

- Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a
**cartesian monoidal category**. For example:**Set**, the category of sets with the Cartesian product, any particular one-element set serving as the unit.**Cat**, the category of small categories with the product category, where the category with one object and only its identity map is the unit.

- Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit. Such a monoidal category is called
**cocartesian monoidal** , the category of modules over a commutative ring*R*-Mod*R*, is a monoidal category with the tensor product of modules ⊗_{R}serving as the monoidal product and the ring*R*(thought of as a module over itself) serving as the unit. As special cases one has:, the category of vector spaces over a field*K*-Vect*K*, with the one-dimensional vector space*K*serving as the unit.**Ab**, the category of abelian groups, with the group of integers**Z**serving as the unit.

- For any commutative ring
*R*, the category of*R*-algebras is monoidal with the tensor product of algebras as the product and*R*as the unit. - The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as the unit.
- The category of all endofunctors on a category
**C**is a*strict*monoidal category with the composition of functors as the product and the identity functor as the unit. - Just like for any category
**E**, the full subcategory spanned by any given object is a monoid, it is the case that for any 2-category**E**, and any object**C**in Ob(**E**), the full 2-subcategory of**E**spanned by {**C**} is a monoidal category. In the case**E**=**Cat**, we get the endofunctors example above. - Bounded-above meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element.
- Any ordinary monoid is a small monoidal category with object set , only identities for morphisms, as tensorproduct and as its identity object. Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product.

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Monoidal preorders, also known as "preordered monoids", are special cases of monoidal categories. This sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. For example, the set of natural numbers has both a monoid structure (using + and 0) and a preorder structure (using ≤), which together form a monoidal preorder, basically because and implies . We now present the general case.

It's well known that a preorder can be considered as a category **C**, such that for every two objects , there exists *at most one* morphism in **C**. If there happens to be a morphism from *c* to *c' *, we could write , but in the current section we find it more convenient to express this fact in arrow form . Because there is at most one such morphism, we never have to give it a name, such as . The reflexivity and transitivity properties of an order are respectively accounted for by the identity morphism and the composition formula in **C**. We write iff and , i.e. if they are isomorphic in **C**. Note that in a partial order, any two isomorphic objects are in fact equal.

Moving forward, suppose we want to add a monoidal structure to the preorder **C**. To do so means we must choose

- an object , called the
*monoidal unit*, and - a functor , which we will denote simply by the dot "", called the
*monoidal multiplication*.

Thus for any two objects we have an object . We must choose and to be associative and unital, up to isomorphism. This means we must have:

- and .

Furthermore, the fact that · is required to be a functor means—in the present case, where **C** is a preorder—nothing more than the following:

- if and then .

The additional coherence conditions for monoidal categories are vacuous in this case because every diagram commutes in a preorder.

Note that if **C** is a partial order, the above description is simplified even more, because the associativity and unitality isomorphisms becomes equalities. Another simplification occurs if we assume that the set of objects is the free monoid on a generating set . In this case we could write , where * denotes the Kleene star and the monoidal unit *I* stands for the empty string. If we start with a set *R* of generating morphisms (facts about ≤), we recover the usual notion of semi-Thue system, where *R* is called the "rewriting rule".

To return to our example, let **N** be the category whose objects are the natural numbers 0, 1, 2, ..., with a single morphism if in the usual ordering (and no morphisms from *i* to *j* otherwise), and a monoidal structure with the monoidal unit given by 0 and the monoidal multiplication given by the usual addition, . Then **N** is a monoidal preorder; in fact it is the one freely generated by a single object 1, and a single morphism 0 ≤ 1, where again 0 is the monoidal unit.

It follows from the three defining coherence conditions that *a large class* of diagrams (i.e. diagrams whose morphisms are built using , , , identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that *all* such diagrams commute.

There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra. Ordinary monoids are precisely the monoid objects in the cartesian monoidal category **Set**. Further, any 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).

Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.

Every monoidal category can be seen as the category **B**(∗, ∗) of a bicategory **B** with only one object, denoted ∗.

A category **C** enriched in a monoidal category **M** replaces the notion of a set of morphisms between pairs of objects in **C** with the notion of an **M**-object of morphisms between every two objects in **C**.

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

- its objects are lists (finite sequences)
*A*_{1}, ...,*A*_{n}of objects of**C**; - there are arrows between two objects
*A*_{1}, ...,*A*_{m}and*B*_{1}, ...,*B*_{n}only if*m*=*n*, and then the arrows are lists (finite sequences) of arrows*f*_{1}:*A*_{1}→*B*_{1}, ...,*f*_{n}:*A*_{n}→*B*_{n}of**C**; - the tensor product of two objects
*A*_{1}, ...,*A*_{n}and*B*_{1}, ...,*B*_{m}is the concatenation*A*_{1}, ...,*A*_{n},*B*_{1}, ...,*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**.

- If, in a monoidal category, and are naturally isomorphic in a manner compatible with the coherence conditions, we speak of a braided monoidal category. If, moreover, this natural isomorphism is its own inverse, we have a symmetric monoidal category.
- A closed monoidal category is a monoidal category where the functor has a right adjoint, which is called the "internal Hom-functor" . Examples include cartesian closed categories such as
**Set**, the category of sets, and compact closed categories such as**FdVect**, the category of finite-dimensional vector spaces. - Autonomous categories (or compact closed categories or rigid categories) are monoidal categories in which duals with nice properties exist; they abstract the idea of
**FdVect**. - Dagger symmetric monoidal categories, equipped with an extra dagger functor, abstracting the idea of
**FdHilb**, finite-dimensional Hilbert spaces. These include the dagger compact categories. - Tannakian categories are monoidal categories enriched over a field, which are very similar to representation categories of linear algebraic groups.

In mathematics, an **associative algebra** is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give *A* the structure of a ring; the addition and scalar multiplication operations together give *A* the structure of a vector space over *K*. In this article we will also use the term *K*-algebra to mean an associative algebra over the field *K*. A standard first example of a *K*-algebra is a ring of square matrices over a field *K*, with the usual matrix multiplication.

In mathematics, the **tensor product***V* ⊗ *W* of two vector spaces *V* and *W* is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product *V* × *W* to *V* ⊗ *W* in a way that generalizes the outer product.

In category theory, a branch of mathematics, an **enriched category** generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint.

In mathematics, the idea of a **free object** is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure. It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices. Informally, a free object over a set *A* can be thought of as being a "generic" algebraic structure over *A*: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

In category theory, a branch of mathematics, a **monad** is an endofunctor, together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

In mathematics, especially in category theory, a **closed monoidal category** is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.

This is a glossary of properties and concepts in category theory in mathematics.

In category theory, a **strict 2-category** is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over **Cat**.

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called **hom-functors** and have numerous applications in category theory and other branches of mathematics.

In mathematics, a **super vector space** is a -graded vector space, that is, a vector space over a field with a given decomposition of subspaces of grade and grade . The study of super vector spaces and their generalizations is sometimes called **super linear algebra**. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry.

In category theory, a branch of mathematics, a **symmetric monoidal category** is a monoidal category such that the tensor product is symmetric. One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field *k,* using the ordinary tensor product of vector spaces.

In mathematics, a ***-autonomous** category **C** is a symmetric monoidal closed category equipped with a dualizing object . The concept is also referred to as **Grothendieck—Verdier category** in view of its relation to the notion of Verdier duality.

In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two *coherence maps*—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

In category theory, a **monoidal monad** is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations and are monoidal natural transformations. In other words, is equipped with coherence maps and satisfying certain properties, and the unit and multiplication are monoidal natural transformations. By monoidality of , the morphisms and are necessarily equal.

In category theory, a branch of mathematics, a **PROP** is a symmetric strict monoidal category whose objects are the natural numbers *n* identified with the finite sets and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each *n*, the symmetric group on *n* letters is given as a subgroup of the automorphism group of *n*. The name PROP is an abbreviation of "PROduct and Permutation category".

In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is **FdVect**, the category having finite-dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is **Rel**, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure.

In category theory, a branch of mathematics, a **dual object** is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a **dualizable object**. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space *V*^{∗} doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property.

In category theory, a branch of mathematics, **dagger compact categories** first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations. They also appeared in the work of John Baez and James Dolan as an instance of semistrict *k*-tuply monoidal *n*-categories, which describe general topological quantum field theories, for *n* = 1 and *k* = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.

In mathematics, **Hochschild homology ** is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).

In category theory, a branch of mathematics, a **rigid category** is a monoidal category where every object is rigid, that is, has a dual *X*^{*} and a morphism **1** → *X* ⊗ *X*^{*} satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined by Neantro Saavedra-Rivano in his thesis on Tannakian categories.

- ↑ Baez, John; Stay, Mike (2011). "Physics, topology, logic and computation: a Rosetta Stone". In Coecke, Bob (ed.).
*New Structures for Physics*. Lecture Notes in Physics.**813**. Springer, Berlin. pp. 95–172. arXiv: 0903.0340 . ISBN 9783642128219. ISSN 0075-8450.

- Joyal, André; Street, Ross (1993). "Braided Tensor Categories".
*Advances in Mathematics**102*, 20–78. - Joyal, André; Street, Ross (1988). "Planar diagrams and tensor algebra".
- Kelly, G. Max (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc."
*Journal of Algebra**1*, 397–402 - Kelly, G. Max (1982).
*Basic Concepts of Enriched Category Theory*(PDF). London Mathematical Society Lecture Note Series No. 64. Cambridge University Press. - Mac Lane, Saunders (1963). "Natural Associativity and Commutativity".
*Rice University Studies**49*, 28–46. - Mac Lane, Saunders (1998),
*Categories for the Working Mathematician*(2nd ed.). New York: Springer-Verlag. - Monoidal category in
*nLab*

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