 Pentagon identity for a monoidal category

ID: pentagon-identity-for-a-monoidal-category

a

in a monoidal category, the two paths from

((W \otimes X) \otimes Y) \otimes Z

W \otimes (X \otimes (Y \otimes Z))

agree:

a_{W, X, Y \otimes Z} a_{W \otimes X, Y, Z} = (1_{W} \otimes a_{X, Y, Z}) a_{W, X \otimes Y, Z} (a_{W, X, Y} \otimes 1_{Z}) .

This five-vertex diagram is the basic compatibility for reassociating four tensor factors. The monoidal coherence theorem extends it to arbitrary finite tensor expressions.

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