This paper introduces a categorical account of infinitesimal causality in Frobenius Markov categories equipped with tangent-bundle semantics. It defines causal sufficiency through the compatibility of two distinct Frobenius structures: one encoding classical variable operations and another representing geometric integrability.
- Infinitesimal interventions are modeled as tangent vectors that deform copy/discard operations, with their Lie brackets measuring preservation of classical information flow.
- Causal sufficiency is defined by the compatibility between the categorical Frobenius algebra on classical variables and the geometric Frobenius integrability condition.
- Pearl's do-calculus serves as a guiding example, linking ignoring irrelevant interventions to counit invariance and independence to involutive bracket closure.
- For structural causal models, infinitesimal causality is formulated in the slice of deterministic mechanisms over exogenous variables, with stochastic kernels appearing only after pushforward.
This framework provides a rigorous mathematical foundation for understanding how interventions act on classical information structures within categorical probability theory.