We construct and study a triangulated category of motives with modulus
over a field
that extends
Voevodsky’s category
in such a way as to encompass nonhomotopy invariant phenomena. In a similar way as
is constructed out
of smooth
-varieties,
is constructed out of
proper modulus pairs, introduced in Part I
of this work. To such a modulus pair we associate its motive in
. In some
cases, the
group in
between the motives of two modulus pairs can be described in terms of Bloch’s higher
Chow groups.