nLab p-connection

For a smooth morphism $p$ of smooth analytic spaces or of smooth schemes $p\colon X \to S$ a $p$-connection is an $\mathcal{O}_X$-linear map $\nabla_S\colon p^* T_S \to T_X$ such that $\mathrm{d}p \circ \nabla_S = id_{p^* T_S}$. The “differential” $\mathrm{d}p$ here is the map $T_X \to p^* T_S$ induced by the universality of the pullback and the differential. A $p$-connection is flat/integrable if the corresponding (by adjunction) map $T_S \to p_* T_X$ commutes with brackets of vector fields.

• Alexander Beilinson, David Kazhdan, Flat projective connections, <http://www.math.sunysb.edu/~kirillov/manuscripts/kazhdan2.pdf>