  We show how the categorial approach to inverse monoids can be described as a
certain endofunctor (which we call the partialization functor) of some
category. In this paper we show that this functor can be used to obtain several
recently defined inverse monoids, and use it to define a new object, which we
call the inverse braid-permutation monoid. A presentation for this monoid is
obtained. Finally, we study some abstract properties of the partialization
functor and its iterations. This leads to a categorification of a monoid of all
order preserving maps, and series of orthodox generalizations of the symmetric
inverse semigroup.
