Monoidal category
In mathematics , a designated monoidal category is a category that double-digit functor and a unit object is equipped.
The link must be associative in the sense that there is a natural equivalence ,
gives; must be left and right neutral in the sense that there are natural equivalences and given by
- and .
These natural transformations are said to be coherent. All necessary coherence conditions follow from the commutativity of the following two diagrams:
and
From these two conditions it follows that every such diagram commutes: This is Mac Lane's "coherence theorem ".
- A monoidal category can be viewed as a bi-category with one object.
- The concept of the monoid object can be defined in a monoidal category, which generalizes that of the monoid .
Examples
Any category containing finite products and an end object can be viewed as a symmetrically monoidal category: the two-digit functor is defined by a natural selection of products and the end object is the unit object. Similarly, we can choose a co- product as the two-digit functor and an initial object as the unit object .
We now show the structure of two such monoidal categories in parallel:
-Mod | set |
---|---|
For a commutative ring the category is -Mod of - modules a symmetric monoidal category with product (the tensor product ) and unity . | The category set is symmetrical monoidal with product and unit . |
A unitary associative algebra is an object of -Mod together with arrows and , for which the following diagrams commute: | A monoid is an object M together with arrows and
, for which the following diagrams commute: |
and | and |
. | . |
A koalgebra is an object C with arrows and , for which the following diagrams commute: | For each object S in the Set category, there are two clearly defined arrows and , for which the following diagrams commute: |
and | and |
. | . |
In particular it is unique because is end object . |
swell
- 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.