Monoidal category

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In mathematics, a monoidal category (or tensor category) is a category \mathbb C equipped with a binary 'tensor' functor \otimes: \mathbb C\times\mathbb C\to\mathbb C and a unit object I. The tensor operation must be associative in the sense that there is a natural isomorphism α with components \alpha_{A,B,C}: (A\otimes B)\otimes C \to A\otimes(B\otimes C); and I must be a left and right identity in the sense that there are natural isomorphisms λ and ρ with components \lambda_A: I\otimes A\to A and \rho_A: A\otimes I\to A respectively.

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 \mathbb C, the diagrams:

Image:Monoidal-category-pentagon.png and Image:Monoidal-category-triangle.png

must commute. It follows from these two conditions that any such diagram commutes: this is Mac Lane's "coherence theorem".

  • A monoidal category may be regarded as a bicategory with one object.
  • Many monoidal categories have additional structure such as braiding or symmetry: the references describe this in detail.
  • There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid.
  • Monoidal categories are used to define models for linear logic.
  • Tensor category theory forms the mathematical foundation for the topological order in condensed matter.

Examples

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.

R-ModSet
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 \nabla:A\otimes A\rightarrow A and \eta: R \rightarrow A satisfying A monoid is an object M together with morphisms \circ: M \times M \rightarrow M and 1: \{*\} \rightarrow M satisfying
Image:R-algebra1.png Image:Monoid1.png
andand
Image:R-algebra2.png. Image:Monoid2.png.
A coalgebra is an object C with morphisms \Delta: C \rightarrow C \otimes C and \epsilon:C\rightarrow R satisfying Any object of Set, S has two unique morphisms \Delta: S \rightarrow S \times S and \epsilon: S \rightarrow \{*\} satisfying
Image:R-coalgebra1.png Image:Comonoid1.png
andand
Image:R-coalgebra2.png. Image:Comonoid2.png.
In particular, ε is unique because { * } is a terminal object.

References

  • 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.

de:Monoidale Kategorie

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