  We propose an infinitesimal counterpart to the notion of braided category.
The corresponding infinitesimal braidings are natural transformations which are
compatible with an underlying braided monoidal structure in the sense that they
constitute a first-order deformation of the braiding. This extends previously
considered infinitesimal symmetric or Cartier categories, where involutivity of
the braiding and an additional commutativity of the infinitesimal braiding with
the symmetry are required. The generalized pre-Cartier framework is then
elaborated in detail for the categories of (co)quasitriangular bialgebra
(co)modules and we characterize the resulting infinitesimal
$\mathcal{R}$-matrices (resp. $\mathcal{R}$-forms) on the bialgebra. It is
proven that the latter are Hochschild $2$-cocycles and that they satisfy an
infinitesimal quantum Yang-Baxter equation, while they are Hochschild
$2$-coboundaries under the Cartier (co)triangular assumption in the presence of
an antipode. We provide explicit examples of infinitesimal braidings,
particularly on quantum $2\times 2$-matrices, $\mathrm{GL}_q(2)$, Sweedler's
Hopf algebra and via Drinfel'd twist deformation. As conceptual tools to
produce examples of infinitesimal braidings we prove an infinitesimal version
of the FRT construction and we provide a Tannaka-Krein reconstruction theorem
for pre-Cartier coquasitriangular bialgebras. We comment on the deformation of
infinitesimal braidings and construct a quasitriangular structure on formal
power series of Sweedler's Hopf algebra.
