Categories by proxy and the limits of Para

Abstract: The notion of parameterization is of great importance in categorical cybernetics, providing space for morphisms to be learnt, or for their choice to be 'externally' determined. At the same time, the concept of 'randomness pushback' tells us that the randomness of a stochastic channel can also (in nice circumstances) be so externalized, leaving instead a random choice of deterministic map. The usual perspective on parameterization is an 'internal' one, treating the parameter as a modification of a morphism's (co)domain. In general, however, this perspective is not wide enough to retain all the structure of the category at hand: an 'external' perspective seems mathematically, as well as philosophically, necessary. (In earlier work, we attempted to provide such an external perspective using an enriched-categorical notion of parameterization, but this is similarly insufficient.)

Here, we describe an alternative perspective, considering an internal category parameterized by its 'external' universe. We build an indexed double category over the double category of spans in the universe, with each base object representing a choice of 'parameterizing context'. When the internal category has limits or a subobject classifier, so does its parameterization; with appropriate quotienting, so does the corresponding Grothendieck construction. By decorating the spans with (sub)distributions, the same facts hold true even in the stochastic case, suggesting semantics for notions of 'stochastic type' and 'stochastic term'. In this setting, we can reformulate Bayesian lenses as "Bayesian dependent optics", treating generative models as such stochastic terms.

category theory

Audience: researchers in the topic

( slides | video )

Intercats: Seminar on Categorical Interaction