  We provide a categorical proof of convergence for martingales and backward
martingales in mean, using enriched category theory. The enrichment we use is
in topological spaces, with their canonical closed monoidal structure, which
encodes a version of pointwise convergence.
  We work in a topologically enriched dagger category of probability spaces and
Markov kernels up to almost sure equality. In this category we can describe
conditional expectations exactly as dagger-split idempotent morphisms, and
filtrations can be encoded as directed nets of split idempotents, with their
canonical partial order structure. As we show, every increasing (or decreasing)
net of idempotents tends topologically to its supremum (or infimum).
  Random variables on a probability space form contravariant functors into
categories of Hilbert and Banach spaces, which we can enrich topologically
using the L^p norms. Martingales and backward martingales can be defined in
terms of these functors. Since enriched functors preserve convergence of nets,
we obtain convergence in the L^p norms. The convergence result for backward
martingales indexed by an arbitrary net, in particular, seems to be new.
  By changing the functor, one can describe more general notions of conditional
expectations and martingales, and if the functor is enriched, one automatically
obtains a convergence result. For instance, one can recover the Bochner-based
notion of vector-valued conditional expectation, and the convergence of
martingales with values in an arbitrary Banach space.
  This work seems to be the first application of topologically enriched
categories to analysis and probability in the literature. We hope that this
enrichment, so often overlooked in the past, will be used in the future to
obtain further convergence results categorically.
