A rig category is a category equipped with two monoidal structures, one distributing over the other. Think of sets and bijections under product and disjoint union, or vector spaces under tensor products and direct sums, or Hilbert spaces and unitary maps with the same operations. Rig structure plays a key role in the study of reversible and quantum programming. We discuss the computational role of the rig structure. Starting from a prop generating a ‘base circuit theory’, we universally extend it to a rig category. We then show that the newly obtained circuit theory is governed by a handful of equations over (multi-)controlled gates. Intuitively, this shows that rig structure can be regarded as controlled monoidal structure. More precisely, this isolates computational control aspects from circuits, which is borne out in many examples, for both Boolean circuits and quantum circuits.

An intuitive feeling for understanding statistics and probability is the formula: Statistical Inference = Probability^-1. Is it possible to translate this intuition into serious mathematics? That is the question that Pugh et al. ask and answer through the theory of Kan extensions. We will unpack how to reinterpret the above formula in a 2-categorical framework by considering the categorification of error minimisation schemes. By choosing a specific context in which to interpret this error (a monoidal preorder), we can leverage this rich theory to guarantee the existence of a global minimisation algorithm. This has potential applications for algorithms in machine learning.