  We develop the foundations of Algebraic Stochastic Calculus, with an aim to
replacing what is typically referred to as Stochastic Calculus by a purely
categorical version thereof. We first give a sheaf theoretic reinterpretation
of Probability Theory. We regard probability spaces (X, F, P) as Grothendieck
sites (F, J_P) on which Brownian motions are defined via sheaves in symmetric
monoidal infinity-categories. Due to the complex nature of such a formalism we
are naturally led to considering a purely categorical, time independent
formalism in which stochastic differential equations are replaced by studying
problems in deformation theory.
