The Category Theory of Causal Models

Building on work of Rubenstein et al, we consider a notion of structurepreserving transformations between structural causal models. We describe a category, in the sense of category theory, of such models, and explore the properties of this category, FinMod. We further generalize this theory to encompass approximate transformations, which only preserve the causal structure up to a certain error, and a larger class of profinite models, which includes for example graphical models on infinite graphs. The language of category theory provides a natural setting for these generalizations, and allows for certain definitions, such as the error of a transformation between infinite models, to be made automatically.