  We develop a theory of enriched categories over a (higher) category M
equipped with a class W of morphisms called homotopy equivalences. We call them
Segal M_W -categories. Our motivation was to generalize the notion of
"up-to-homotopy monoids" in a monoidal category M, introduced by Leinster. The
formalism adopted generalizes the classical Segal categories and extends the
theory of enriched category over a bicategory. In particular we have a linear
version of Segal categories which did not exist so far. Our goal in this paper
is to present the theory and provide some examples. Applications are reserved
for the future.
