Monoidal category

Last updated October 01, 2024

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

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, for which monoidal categories can be considered an abstraction, is a system of data types closed under a type constructor that takes two types and builds an aggregate type. The types serve as 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 $((a,b),c)$ and $(a,(b,c))$ —store the same information even though the aggregate values need not be the same. The aggregate type may be analogous to the operation of addition (type sum) or of multiplication (type product). For type product, the identity object is the unit $()$ , 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 it is impossible to address an inhabitant. 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 physics. Braided monoidal categories have applications in quantum information, quantum field theory, and string theory.

Formal definition

A monoidal category is a category $\mathbf {C}$ equipped with a monoidal structure. A monoidal structure consists of the following:

a bifunctor $\otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C}$ called the monoidal product,^[2] or tensor product ,
an object $I$ called the monoidal unit,^[2]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 $A$ , $B$ , $C$ ) isomorphism $\alpha$ , called associator, with components $\alpha _{A,B,C}\colon A\otimes (B\otimes C)\cong (A\otimes B)\otimes C$ ,
- has $I$ as left and right identity: there are two natural isomorphisms $\lambda$ and $\rho$ , respectively called left and right unitor, with components $\lambda _{A}\colon I\otimes A\cong A$ and $\rho _{A}\colon A\otimes I\cong A$ .

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

The coherence conditions for these natural transformations are:

for all $A$ , $B$ , $C$ and $D$ in $\mathbf {C}$ , the pentagon diagram

commutes;

for all $A$ and $B$ in $\mathbf {C}$ , the triangle diagram

This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects. Monoidal2.svg — This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects.

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.

Examples

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
R-Mod, the category of modules over a commutative ring 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:
- K-Vect, the category of vector spaces over a field 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 $(M,\cdot ,1)$ is a small monoidal category with object set $M$ , only identities for morphisms, $\cdot$ as tensor product and $1$ 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.
Any commutative monoid $(M,\cdot ,1)$ can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid. By an Eckmann-Hilton argument, adding another monoidal product on $M$ requires the product to be commutative.

Properties and associated notions

It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using $\alpha$ , $\lambda$ , $\rho$ , 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 (small) 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 ∗.

The concept of 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.

Free strict monoidal category

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.

Specializations

If, in a monoidal category, $A\otimes B$ and $B\otimes A$ 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 $X\mapsto X\otimes A$ has a right adjoint, which is called the "internal Hom-functor" $X\mapsto \mathrm {Hom} _{\mathbf {C} }(A,X)$ . 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.

Preordered monoids

A preordered monoid is a monoidal category in which for every two objects $c,c'\in \mathrm {Ob} (\mathbf {C} )$ , there exists at most one morphism $c\to c'$ in C. In the context of preorders, a morphism $c\to c'$ is sometimes notated $c\leq c'$ . The reflexivity and transitivity properties of an order, defined in the traditional sense, are incorporated into the categorical structure by the identity morphism and the composition formula in C, respectively. If $c\leq c'$ and $c'\leq c$ , then the objects $c,c'$ are isomorphic which is notated $c\cong c'$ .

Introducing a monoidal structure to the preorder C involves constructing

an object $I\in \mathbf {C}$ , called the monoidal unit, and
a functor $\mathbf {C} \times \mathbf {C} \to \mathbf {C}$ , denoted by " $\;\cdot \;$ ", called the monoidal multiplication.

$I$ and $\cdot$ must be unital and associative, up to isomorphism, meaning:

(c_{1}\cdot c_{2})\cdot c_{3}\cong c_{1}\cdot (c_{2}\cdot c_{3})

and

I\cdot c\cong c\cong c\cdot I

.

As · is a functor,

if

c_{1}\to c_{1}'

and

c_{2}\to c_{2}'

then

(c_{1}\cdot c_{2})\to (c_{1}'\cdot c_{2}')

.

The other coherence conditions of monoidal categories are fulfilled through the preorder structure as every diagram commutes in a preorder.

The natural numbers are an example of a monoidal preorder: having both a monoid structure (using + and 0) and a preorder structure (using ≤) forms a monoidal preorder as $m\leq n$ and $m'\leq n'$ implies $m+m'\leq n+n'$ .

The free monoid on some generating set produces a monoidal preorder, producing the semi-Thue system.

Related Research Articles

In mathematics, an associative algebraA over a commutative ring K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication. 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 module or vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over K. A standard first example of a K-algebra is a ring of square matrices over a commutative ring K, with the usual matrix multiplication.

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, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. 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. Examples include free groups, tensor algebras, or free lattices.

In category theory, a branch of mathematics, a monad is a triple $consisting of a functor T from a category to itself and two natural transformations that satisfy the conditions like associativity. For example, if are functors adjoint to each other, then together with determined by the adjoint relation is a monad.$

In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.

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.

In mathematics, a commutativity constraint $on a monoidal category is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have for all pairs of objects .$

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

In mathematics, specifically in category theory, hom-sets 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, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms

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 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, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.

In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category.

References

↑ Baez, John; Stay, Mike (2011). "Physics, topology, logic and computation: a Rosetta Stone" (PDF). In Coecke, Bob (ed.). New Structures for Physics. Lecture Notes in Physics. Vol. 813. Springer. pp. 95–172. arXiv: 0903.0340 . CiteSeerX 10.1.1.296.1044 . doi:10.1007/978-3-642-12821-9_2. ISBN 978-3-642-12821-9. ISSN 0075-8450. S2CID 115169297. Zbl 1218.81008.
1 2 Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv: 1803.05316 [math.CT].

Joyal, André; Street, Ross (1993). "Braided Tensor Categories" (PDF). Advances in Mathematics . 102 (1): 20–78. doi: 10.1006/aima.1993.1055 .
Joyal, André; Street, Ross (1988). "Planar diagrams and tensor algebra" (PDF).
Kelly, G. Max (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc". Journal of Algebra. 1 (4): 397–402. doi: 10.1016/0021-8693(64)90018-3 .
Kelly, G.M. (1982). Basic Concepts of Enriched Category Theory (PDF). London Mathematical Society Lecture Note Series. Vol. 64. Cambridge University Press. ISBN 978-0-521-28702-9. OCLC 1015056596. Zbl 0478.18005.
Mac Lane, Saunders (1963). "Natural Associativity and Commutativity". Rice University Studies. 49 (4): 28–46. CiteSeerX 10.1.1.953.2731 . hdl:1911/62865.
Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.
Perrone, Paolo (2024). "Chapter 6. Monoidal categories". Starting Category Theory. World Scientific. doi:10.1142/9789811286018_0005. ISBN 978-981-12-8600-1.
Selinger, P. (2010). "A Survey of Graphical Languages for Monoidal Categories". New Structures for Physics. Lecture Notes in Physics. Vol. 813. pp. 289–355. arXiv: 0908.3347 . doi:10.1007/978-3-642-12821-9_4. ISBN 978-3-642-12820-2.
Monoidal category at the nLab

External links

Media related to Monoidal category at Wikimedia Commons

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Baez, John; Stay, Mike (2011). "Physics, topology, logic and computation: a Rosetta Stone" (PDF). In Coecke, Bob (ed.). New Structures for Physics. Lecture Notes in Physics. Vol. 813. Springer. pp. 95–172. arXiv: 0903.0340 . CiteSeerX 10.1.1.296.1044 . doi:10.1007/978-3-642-12821-9_2. ISBN 978-3-642-12821-9. ISSN 0075-8450. S2CID 115169297. Zbl 1218.81008.

[seven-sketches-2] 1 2 Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv: 1803.05316 [math.CT].

[1]

[2]