Algebras and modules
Model category presentations
Geometry on formal duals of algebras
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Higher category theory
higher category theory
Extra properties and structure
In algebra, given any non-associative algebra , then the trilinear map
given on any elements by
is called the associator (in analogy with the commutator ).
An algebra for which the associator vanishes is hence an associative algebra. If the associator is possibly non-vanishing but completely anti-symmetric (in that for any permutation of three elements then for the signature of the permutation) then is called an alternative algebra.
In category theory and higher category theory (for monoidal categories, bicategories and their higher versions) one considers relaxing the equation that exhibits the vanishing of the associator
to a natural equivalence
By slight mismatch with the terminology in algebra, it is then this equivalence which is called the associator.
In a bicategory the composition of 1-morphisms does not satisfy associativity as an equation, but there are natural associator 2-morphisms
that satisfy a coherence law among themselves.
If one thinks of the bicategory as obtained from a geometrically defined 2-category , then the composition opeeration of 1-morphisms is a choise of 2-horn-fillers and the associator is a choice of filler of the spheres formed by these.
In monoidal categories
By the periodic table of higher categories a monoidal category is a pointed bicategory with a single object, its objects are the 1-morphisms of the bicategory. Accordingly, here the associator is a natural isomorphism
relating the triple tensor products of these objects.