Applied category theory has recently developed libraries for computing with morphisms in interesting categories, while machine learning has developed ways of learning programs in interesting languages. Taking the analogy between categories and languages seriously, this paper defines a probabilistic generative model of morphisms in free monoidal categories over domain-specific generating objects and morphisms. The paper shows how acyclic directed wiring diagrams can model specifications for morphisms, which the model can use to generate morphisms. Amortized variational inference in the generative model then enables learning of parameters (by maximum likelihood) and inference of latent variables (by Bayesian inversion). A concrete experiment shows that the free category prior achieves competitive reconstruction performance on the Omniglot dataset.

Humans surpass the cognitive abilities of most other animals in our ability to "chunk" concepts into words, and then combine the words to combine the concepts. In this process, we make "infinite use of finite means", enabling us to learn new concepts quickly and nest concepts within each-other. While program induction and synthesis remain at the heart of foundational theories of artificial intelligence, only recently has the community moved forward in attempting to use program learning as a benchmark task itself. The cognitive science community has thus often assumed that if the brain has simulation and reasoning capabilities equivalent to a universal computer, then it must employ a serialized, symbolic representation. Here we confront that assumption, and provide a counterexample in which compositionality is expressed via network structure: the free category prior over programs. We show how our formalism allows neural networks to serve as primitives in probabilistic programs. We learn both program structure and model parameters end-to-end.