monoidal category

A monoidal category is a categoryMathworldPlanetmath which has the structureMathworldPlanetmath of a monoid, that is, among the objects there is a binary operationMathworldPlanetmath which is associative and has an unique neutral or unit element. Specifically, a category 𝒞 is monoidal if

  1. 1.

    there is a bifunctor :𝒞×𝒞𝒞, where the images of object (A,B) and morphismMathworldPlanetmath (f,g) are written AB and fg respectively,

  2. 2.

    there is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath aABC:(AB)CA(BC), for arbitrary objects A,B,C in 𝒞, such that aABC is natural in A,B and C. In other words,

    • a-BC:(-B)C-(BC) is a natural transformation for arbitrary objects B,C in 𝒞,

    • aA-C:(A-)CA(-C) is a natural transformation for arbitrary objects A,C in 𝒞,

    • aAB-:(AB)-A(B-) is a natural transformation for arbitrary objects A,B in 𝒞,

  3. 3.

    there is an object I in 𝒞 called the unit object (or simply the unit),

  4. 4.

    for any object A in 𝒞, there are isomorphisms:

    lA:IAA  and  rA:AIA,

    such that lA and rA are natural in A: both l:I-- and r:-I- are natural transformations

satisfying the following commutative diagramsMathworldPlanetmath:

  • unit coherence law

  • associativity coherence law


The bifunctor is called the tensor productPlanetmathPlanetmath on 𝒞, and the natural isomorphisms a,l,r are called the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.

Some examples of monoidal categories are

  • A prototype is the category of isomorphism classes of vector spaces over a field 𝕂, herein the tensor product is the associative operationMathworldPlanetmath and the field 𝕂 itself is the unit element.

  • The category of sets is monoidal. The tensor product here is just the set-theoretic cartesian productMathworldPlanetmath, and any singleton can be used as the unit object.

  • The category of (left) modules over a ring R is monoidal. The tensor product is the usual tensor product ( of modules, and R itself is the unit object.

  • The category of bimodules over a ring R is monoidal. The tensor product and the unit object are the same as in the previous example.

Monoidal categories play an important role in the topological quantum field theories (TQFT).

Title monoidal category
Canonical name MonoidalCategory
Date of creation 2013-03-22 16:30:21
Last modified on 2013-03-22 16:30:21
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 14
Author juanman (12619)
Entry type Definition
Classification msc 81-00
Classification msc 18-00
Classification msc 18D10
Synonym monoid
Related topic Category
Related topic Algebroids
Related topic Monoid
Related topic StateOnTheTetrahedron
Defines unit coherence
Defines associativity coherence
Defines tensor product
Defines unit object
Defines associativity isomorphism
Defines left unit isomorphism
Defines right unit isomorphism