  Interventional causal models describe joint distributions over some variables
used to describe a system, one for each intervention setting. They provide a
formal recipe for how to move between joint distributions and make predictions
about the variables upon intervening on the system. Yet, it is difficult to
formalise how we may change the underlying variables used to describe the
system, say from fine-grained to coarse-grained variables. Here, we argue that
compositionality is a desideratum for model transformations and the associated
errors. We develop a framework for model transformations and abstractions with
a notion of error that is compositional: when abstracting a reference model M
modularly, first obtaining M' and then further simplifying that to obtain M'',
then the composite transformation from M to M'' exists and its error can be
bounded by the errors incurred by each individual transformation step. Category
theory, the study of mathematical objects via the compositional transformations
between them, offers a natural language for developing our framework. We
introduce a category of finite interventional causal models and, leveraging
theory of enriched categories, prove that our framework enjoys the desired
compositionality properties.
