# Monoidal Equivalence: Two Definitions

One has the definition of a monoidal equivalence as in definition 12 of Baez's Some Definitions Everyone Should Know.

I have also seen monoidal equivalence defined as a monoidal functor between monoidal categories which defines an equivalence (i.e. a fully faithful, essentially surjective functor) in the sense of definition 5 in Baez (link above).

I wanted to double-check that these two definitions are distinct. If anybody has any intuitive insight as to why/when the two are employed, that would also be appreciated. Thanks.

• According to the comments on this question, they are actually equivalent : math.stackexchange.com/questions/2563128/… – Arnaud D. Jun 12 '18 at 8:47
• A priori they're distinct but I wouldn't be surprised to find out they're the same (like the two definitions of equivalence) – Max Jun 12 '18 at 8:55
• Thanks, according to the discussion in the link above the two are equivalent. To show it apparently isn't difficult, yet requires quite some time and effort. An official source vouching for their equivalence is Remark 2.4.10 in www-math.mit.edu/~etingof/egnobookfinal.pdf – S Valera Jun 12 '18 at 8:57