What is a monoidal groupoid?

Question

I have heard the term monoidal groupoid mentioned, and I'm trying to understand what it means.

I understand what a group is, and that a monoid is similar to a group except that each element need not have an inverse. Also, it seems that a groupoid is like a group where different elements may act on different (isomorphic) underlying objects.

Combining these definitions, it seems as if a monoidal groupoid should be a set of (potentially non-invertible) morphisms between isomorphic objects. So it is essentially a category, but with the additional restriction that the objects of the category be isomorphic. Is this correct?

Answer 1

No. "Monoidal groupoid" is a combination of "groupoid" with "monoidal category", not "monoid". So a monoidal groupoid is a groupoid (category in which every morphism is invertible) which is also equipped with the structure of a monoidal category.

Note also that not all objects in a groupoid need to be isomorphic. All the morphisms are isomorphisms, but some pairs of objects might have no morphisms between them at all.

Answer 2

A monoidal groupoid is just a monoidal category that is also a groupoid.

In other words, it's a monoidal category where all morphisms are isomorphisms.