There is a substantial theory (modelled on permutation representations
of groups) of representations of an inverse semigroup S in a symmetric
inverse monoid I_{X}, that is, a monoid of partial one-to-one
selfmaps of a set X. The present paper describes the structure of a
categorical dual I*_{X} to the symmetric inverse monoid and
discusses representations of an inverse semigroup in this dual symmetric
inverse monoid. It is shown how a representation of S by (full)
selfmaps of a set X leads to dual pairs of representations in
I_{X} and I*_{X}, and how a number of known
representations arise as one or the other of these pairs. Conditions on S
are described which ensure that representations of S preserve such infima
or suprema as exist in the natural order of S. The categorical treatment
allows the construction, from standard functors, of representations of S
in certain other inverse algebras (that is, inverse monoids in which all
finite infima exist). The paper concludes by distinguishing two
subclasses of inverse algebras on the basis of their embedding
properties.