  The aim of this work is to construct certain homotopy t-structures on various
categories of motivic homotopy theory, extending works of Voevodsky, Morel,
D\'eglise and Ayoub. We prove these $t$-structures possess many good
properties, some analogous to that of the perverse $t$-structure of Beilinson,
Bernstein and Deligne. We compute the homology of certain motives, notably in
the case of relative curves. We also show that their hearts provide convenient
extensions of the theory of homotopy invariant sheaves with transfers,
extending some of the main results of Voevodsky. These t-structures are closely
related to Gersten weight structures as defined by Bondarko.
