Learners are Almost Free Compact Closed

The category of learners has a pleasant symmetric formulation when the morphisms are considered up to a coarser equivalence than the one originally described in the paper "Backprop as Functor".A quotient of this modified category gives a new construction of the free compact closed category on a symmetric monoidal category.In this note I expand on an observation made at the end of [15] and further discussed online [16].If the definition of a learner [7] is tweaked slightly so that learners are considered identical under a coarser equivalence relation than before, a striking symmetry in the definition is revealed.This new category comes quite close to being compact closed.There is a strict symmetric monoidal involution and each object has a cup and cap morphism connecting it to its "dual" under this involution.However, the snake equations fail and so these are not true duals.Quotienting the category to force the equations to hold results in a compact closed category, and we end the note by showing that it is in fact the free compact closed category on the starting symmetric monoidal category.Compact closed categories support a more powerful diagrammatic calculus than bare symmetric monoidal categories, so this work may make it easier to design and reason about learners compositionally.The operation that dualises a learner does not appear to have been previously noticed and there is much to investigate.As asked by David Spivak, what is the dual of a neuron?