Locally commutative categories

It is known that a finite category can have all its base monoids in a variety V (i.e. be locally V, denoted V) without itself dividing a monoid in V (i.e. be globally V, denoted gV). This is in particular the case when V=Com, the variety of commutative monoids. Our main result provides a combinatorial characterization of locally commutative categories. This is the first such theorem dealing with a variety for which local differs from global. As a consequence, we show that lCom subset of gV for every variety V that strictly contains the commutative monoids.