  Locally inverse semigroups are regular semigroups whose idempotents form
pseudo-semilattices. We characterise the categories that correspond to locally
inverse semigroups in the realm of Nambooripad's cross-connection theory.
Further, we specialise our cross-connection description of locally inverse
semigroups to inverse semigroups and completely 0-simple semigroups, obtaining
structure theorems for these classes. In particular, we show that the structure
theorem for inverse semigroups can be obtained using only one category, quite
analogous to the Ehresmann-Schein-Nambooripad Theorem; for completely 0-simple
semigroups, we show that cross-connections coincide with structure matrices,
thus recovering the Rees Theorem by categorical tools.
