  The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for
inverse monoids: they are inductive groupoids. A particularly nice case due to
Jarek is that commutative inverse monoids become semilattices of abelian
groups. It has also been categorified by DeWolf-Pronk to a structure theorem
for inverse categories as locally complete inductive groupoids. We show that in
the case of compact inverse categories, this takes the particularly nice form
of a semilattice of compact groupoids. Moreover, one-object compact inverse
categories are exactly commutative inverse monoids. Compact groupoids, in turn,
are determined in particularly simple terms of 3-cocycles by Baez-Lauda.
