  A locally finite, causal and quantal substitute for a locally Minkowskian
principal fiber bundle $\cal{P}$ of modules of Cartan differential forms $\omg$
over a bounded region $X$ of a curved $C^{\infty}$-smooth differential manifold
spacetime $M$ with structure group ${\bf G}$ that of orthochronous Lorentz
transformations $L^{+}:=SO(1,3)^{\uparrow}$, is presented. ${\cal{P}}$ is the
structure on which classical Lorentzian gravity, regarded as a Yang-Mills type
of gauge theory of a $sl(2,\com)$-valued connection 1-form $\cal{A}$, is
usually formulated. The mathematical structure employed to model this
replacement of ${\cal{P}}$ is a principal finitary spacetime sheaf
$\vec{\cal{P}}_{n}$ of quantum causal sets $\amg_{n}$ with structure group
${\bf G}_{n}$, which is a finitary version of the group ${\bf G}$ of local
symmetries of General Relativity, and a finitary Lie algebra ${\bf
g}_{n}$-valued connection 1-form ${\cal{A}}_{n}$ on it, which is a section of
its sub-sheaf $\amg^{1}_{n}$. ${\cal{A}}_{n}$ is physically interpreted as the
dynamical field of a locally finite quantum causality, while its associated
curvature ${\cal{F}}_{n}$, as some sort of `finitary Lorentzian quantum
gravity.
