  Cross-connection theory developed by Nambooripad is the construction of a
semigroup from its principal left (right) ideals using categories. We briefly
describe the general cross-connection theory for regular semigroups and use it
to study the {normal categories} arising from the semigroup $Sing(V)$ of
singular linear transformations on an arbitrary vectorspace $V$ over a field
$K$. There is an inbuilt notion of duality in the cross-connection theory, and
we observe that it coincides with the conventional algebraic duality of vector
spaces. We describe various cross-connections between these categories and show
that although there are many cross-connections, upto isomorphism, we have only
one semigroup arising from these categories. But if we restrict the categories
suitably, we can construct some interesting subsemigroups of the {variant} of
the linear transformation semigroup.
