  We present a framework for characterizing injectivity of classes of maps (on
cosets of a linear subspace) by injectivity of classes of matrices. Using our
formalism, we characterize injectivity of several classes of maps, including
generalized monomial and monotonic (not necessarily continuous) maps. In fact,
monotonic maps are special cases of {\em component-wise affine} maps. Further,
we study compositions of maps with a matrix and other composed maps, in
particular, rational functions. Our framework covers classical injectivity
criteria based on mean value theorems for vector-valued maps and recent results
obtained in the study of chemical reaction networks.
