Uncertainty in category theory

What does category theory have to say about missing data, incomplete information, and other forms of uncertainty?

Literature

General

  • Johnson & Rosebrugh, 2003: Three approaches to partiality in the sketch data model (doi, pdf)
  • Marsden, 2016: Ambiguity and incomplete information in categorical models of language (arxiv, pdf, slides)
    • General strategy:
      1. Define a commutative monad \(T\) on \(\mathbf{Set}\)
      2. Its Eilenberg-Moore algebras form a bicomplete, closed symmetric monoidal category \(\mathrm{EM}(T)\)
      3. Take categories enriched in \(\mathrm{EM}(T)\), usually generated freely from an existing category like \(\mathbf{Rel}\) or \(\mathbf{FdHilb}\)
    • Includes both qualitative and quantitative models
    • The qualitative models are not very convincing

Probabilistic

TODO