A normal band is a semigroup B satisfying a 2 = a and a b c a = a c b a for all a , b , c ∈ B . Normal bands are characterized as strong semilattices of rectangular bands. Normal categories are abstractions of the category of principal left (or right) ideals of regular semigroups. We characterize the category of principal left ideals of a normal band as an N B -category and describe a special cross connection which provides a normal band as the cross connection semigroup. The morphisms in the normal category L (B) arising from a normal band B admit a partial order which is induced by the natural partial order on B. We introduce a partial order on the morphisms of an N B -category which behaves like the partial order on L (B) . In particular, we show that each homset contains a maximum element. Using this partial order, we get a simplified description of the cross connection semigroup. [ABSTRACT FROM AUTHOR]