nLab
associator

• associator , unitor, Jacobiator

• pentagonator, hexagonator

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Idea

An associator in category theory and higher category theory is an isomorphism that relaxes the ordinary associativity equality of a binary operation.

In Bicategories

In a bicategory the composition of 1-morphisms does not satisfy associativity as an equation, but there are natural associator 2-morphisms

$$h\circ (g\circ f)\stackrel{\simeq }{\Rightarrow }(h\circ g)\circ f$$h \circ (g \circ f) \stackrel{\simeq}{\Rightarrow} (h \circ g) \circ f

that satisfy a coherence law among themselves.

If one thinks of the bicategory as obtained from a geometrically defined 2-category $C$, 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 $\partial \Delta [3]\to C$ 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

$$a_{x,y,z}:(x\otimes y)\otimes z\to x(\otimes y\otimes z)$$a_{x,y,z} : (x \otimes y) \otimes z \to x (\otimes y \otimes z)

relating the triple tensor products of these objects.