Monoidal categories

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higher category theory

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1-categorical presentations



In algebra, given any non-associative algebra AA, then the trilinear map

[,,]AAAA [-,-,-] \;-\; A \otimes A \otimes A \longrightarrow A

given on any elements a,b,cAa,b,c \in A by

[a,b,c](ab)ca(bc) [a,b,c] \coloneqq (a b) c - a (b c)

is called the associator (in analogy with the commutator [a,b]abba[a,b] \coloneqq a b - b a ).

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 σ\sigma of three elements then [a σ 1,a σ 2,a σ 3]=(1) |σ|[a 1,a 2,a 3][a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3] for |σ|\vert \sigma \vert the signature of the permutation) then AA 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

(ab)c=a(bc) (a b) c = a (b c)

to a natural equivalence

α a,b,c:(ab)ca(bc). \alpha_{a,b,c} \;\colon\; (a b) c \overset{\simeq}{\longrightarrow} a (b c) \,.

By slight mismatch with the terminology in algebra, it is then this equivalence which is called the associator.

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(gf)(hg)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 CC, 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 Δ[3]C\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:(xy)zx(yz) a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z)

relating the triple tensor products of these objects.


Revised on January 4, 2017 07:56:41 by Urs Schreiber (