monoidal category (original) (raw)

A monoidal category is a category which has the structure of a monoid, that is, among the objects there is a binary operation 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×𝒞→𝒞, where the images of object (A,B) and morphism (f,g) are written A⊗B and f⊗g respectively,
2. 1. there is an isomorphism aA⁢B⁢C:(A⊗B)⊗C≅A⊗(B⊗C), for arbitrary objects A,B,C in 𝒞, such that aA⁢B⁢C is natural in A,B and C. In other words,
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     aA-C:(A⊗-)⊗C⇒A⊗(-⊗C) is a natural transformation for arbitrary objects A,C in 𝒞,
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     aA⁢B-:(A⊗B)⊗-⇒A⊗(B⊗-) is a natural transformation for arbitrary objects A,B in 𝒞,
3. 1. there is an object I in 𝒞 called the unit object (or simply the unit),
4. 1. for any object A in 𝒞, there are isomorphisms:
      such that lA and rA are natural in A: both l:I⊗-⇒- and r:-⊗I⇒- are natural transformations

satisfying the following commutative diagrams:

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  unit coherence law

  \xymatrix⁢@+=2⁢c⁢m⁢(A⊗I)⊗B⁢\ar⁢[r⁢r]aA⁢I⁢B⁢\ar⁢[d⁢r]rA⊗1B⁢&⁢&⁢A⊗(I⊗B)⁢\ar⁢[d⁢l]1A⊗rB⁢&⁢A⊗B⁢&

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  associativity coherence law

  \xymatrix⁢@+=2⁢c⁢m⁢((A⊗B)⊗C)⊗D⁢\ar⁢[r⁢r]aA⊗B,C,C⁢\ar⁢[d]aA⁢B⁢C⊗1D⁢&⁢&⁢(A⊗B)⊗(C⊗D)⁢\ar⁢[d⁢d]aA,B,C⊗D⁢(A⊗(B⊗C))⊗D⁢\ar⁢[d]aA,B⊗C,D⁢&⁢&⁢A⊗((B⊗C)⊗D)⁢\ar⁢[r⁢r]1A⊗aB⁢C⁢D⁢&⁢&⁢A⊗(B⊗(C⊗D))

The bifunctor ⊗ is called the tensor product 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

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  A prototype is the category of isomorphism classes of vector spaces over a field 𝕂, herein the tensor product is the associative operation and the field 𝕂 itself is the unit element.
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  The category of (left) modules over a ring R is monoidal. The tensor product is the usual tensor product (http://planetmath.org/TensorProduct) of modules, and R itself is the unit object.
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  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).