Monoidal category

From Free net encyclopedia

(Redirected from Algebra/set analogy)

In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). More explicitly, a monoidal category is a category <math>\mathbb C</math> equipped with a binary functor

<math>\otimes: \mathbb C\times\mathbb C\to\mathbb C</math>

called tensor, and a unit object <math>I</math>.

A monoidal category must be equipped with three natural isomorphisms expressing the fact that the tensor operation should

  • be associative: there is a natural isomorphism <math>\alpha</math>, called associativity, with components
<math>\alpha_{A,B,C}: (A\otimes B)\otimes C \to A\otimes(B\otimes C)</math>,
  • have <math>I</math> as left and right identity: there are two natural isomorphisms <math>\lambda</math> and <math>\rho</math>, respectively called left and right identity, with components
<math>\lambda_A: I\otimes A\to A</math>

and

<math>\rho_A: A\otimes I\to A</math>.

These natural transformations are subject to certain coherence conditions. All the necessary conditions are implied by the following two: for all <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> in <math>\mathbb C</math>, 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 (i.e. a diagram whose morphisms are built using <math>\alpha</math>, <math>\lambda</math>, <math>\rho</math>, identities and tensor product) commutes: this is Mac Lane's "coherence theorem". This can be equivalently formulated by saying that every monoidal category is equivalent to a strict (see below) monoidal category.

Monoidal categories are used to define models for the multiplicative fragment of intuitionistic linear logic. They also from the mathematical foundation for the topological order in condensed matter.

Contents

[edit]

Lax monoidal categories

A monoidal category is said to be a strict monoidal category when the natural isomorphisms <math>\alpha</math>, <math>\lambda</math> and <math>\rho</math> are identities. A lax monoidal category is a generalization of the notion of monoidal category where the natural transformations <math>\alpha</math>, <math>\lambda</math> and <math>\rho</math> are not required to be isomorphisms. A monoidal category is sometimes called a strong monoidal cateory or a weak monoidal category to emphasize that it is not lax.

A lax monoidal category may be regarded as a bicategory with one object.

[edit]

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 <math>R</math>-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.

<math>R</math>-ModSet
Given a field or commutative ring <math>R</math>, the category <math>R</math>-Mod of <math>R</math>-modules (in the case of a field, vector spaces) is a symmetric monoidal category with product ⊗ and identity <math>R</math>. The category Set is a symmetric monoidal category with product × and identity {*}.
A unital associative algebra is an object of <math>R</math>-Mod together with morphisms <math>\nabla:A\otimes A\rightarrow A</math> and <math>\eta: R \rightarrow A</math> satisfying A monoid is an object M together with morphisms <math>\circ: M \times M \rightarrow M</math> and <math>1: \{*\} \rightarrow M</math> satisfying
Image:R-algebra1.png Image:Monoid1.png
andand
Image:R-algebra2.png. Image:Monoid2.png.
A coalgebra is an object C with morphisms <math>\Delta: C \rightarrow C \otimes C</math> and <math>\epsilon:C\rightarrow R</math> satisfying Any object of Set, S has two unique morphisms <math>\Delta: S \rightarrow S \times S</math> and <math>\epsilon: S \rightarrow \{*\}</math> satisfying
Image:R-coalgebra1.png Image:Comonoid1.png
andand
Image:R-coalgebra2.png. Image:Comonoid2.png.
In particular, ε is unique because <math>\{*\}</math> is a terminal object.
[edit]

See also

  • Many monoidal categories have additional structure such as braiding, symmetry or closure: the references describe this in detail.
  • Monoidal functors are the functors between monoidal categories which preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.
  • There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid. In particular, a 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).
[edit]

References

Retrieved from "http://www.netipedia.com/index.php/Monoidal_category"