  We propose a mathematical framework for a unification of the distributional
theory of meaning in terms of vector space models, and a compositional theory
for grammatical types, for which we rely on the algebra of Pregroups,
introduced by Lambek. This mathematical framework enables us to compute the
meaning of a well-typed sentence from the meanings of its constituents.
Concretely, the type reductions of Pregroups are `lifted' to morphisms in a
category, a procedure that transforms meanings of constituents into a meaning
of the (well-typed) whole. Importantly, meanings of whole sentences live in a
single space, independent of the grammatical structure of the sentence. Hence
the inner-product can be used to compare meanings of arbitrary sentences, as it
is for comparing the meanings of words in the distributional model. The
mathematical structure we employ admits a purely diagrammatic calculus which
exposes how the information flows between the words in a sentence in order to
make up the meaning of the whole sentence. A variation of our `categorical
model' which involves constraining the scalars of the vector spaces to the
semiring of Booleans results in a Montague-style Boolean-valued semantics.
