  The notions of $p$-convexity and $q$-concavity are mostly known because of
their importance as a tool in the study of isomorphic properties of Banach
lattices, but they also play a role in several results involving linear maps
between Banach spaces and Banach lattices. In this paper we introduce Lipschitz
versions of these concepts, dealing with maps between metric spaces and Banach
lattices, and start by proving nonlinear versions of two well-known
factorization theorems through $L_p$ spaces due to Maurey/Nikishin and Krivine.
We also show that a Lipschitz map from a metric space into a Banach lattice is
Lipschitz $p$-convex if and only if its linearization is $p$-convex.
Furthermore, we elucidate why there is such a close relationship between the
linear and nonlinear concepts by proving characterizations of Lipschitz
$p$-convex and Lipschitz $q$-concave maps in terms of factorizations through
$p$-convex and $q$-concave Banach lattices, respectively, in the spirit of the
work of Raynaud and Tradacete.
