Example [efr-TO9K]

Given a compact Hausdorff space X, a Banach space bundle is a space over X, V \to X, equipped with a fiberwise (complex) vector space structure +: V \times _X V \to V, \cdot : \mathbb {C} \times V \to V, so that there exists a cover \{U_i\} of X so that for each U_i, there exists a Banach space V_i and a homeomorphism V \times _X U_i =: V_{U_i} \cong V_i \times U_i over U_i, which is moreover linear in each fiber, where V_i is equipped with the norm topology.

Note that this determines a local norm on each V_{U_i} (and in particular each V_x) up to equivalence (but no stricter than that). In particular each V_x is a Banach space.

A morphism of Banach space bundles is a continuous map f: V \to W over X which is linear on each bundle. Note that this implies that on a suitable cover U_i, the maps f: V_{U_i} \to W_{U_i} obey \left \lVert f(v) \right \rVert \leq C_i \left \lVert v \right \rVert for some C_i \in \mathbb {R}, for any local norms inducing the topologies, and hence by compactness there exists some C so that \left \lVert f(v) \right \rVert \leq C \left \lVert v \right \rVert for each v.

If X \to Y is a continuous map, there is a pullback functor \mathsf {Ban}_Y \to \mathsf {Ban}_X. The Grothendieck construction of this gives a fibration \mathsf {BanBun} \to \mathsf {CHaus}

Note that Tychonoff spaces include all compact Hausdorff spaces. Therfore consider the full subcategory \mathsf {CHausStoch} \hookrightarrow \mathsf {TychStoch} spanned by these. The fibration \mathsf {BanBun} admits the structure of a stochastic module: given M \to X, s: X \to M a kernel, and a linear continuous map f: V \times _X M \to W \times _X M, given v \in V_x, there is an induced function M_x \to W_x given by f(v,-). Since this is bounded (being continuous on a compact space) it is (Bochner) integrable, define s^*f(v) to be this integral.

This example is analogous to optics for the action of categories of markov kernels on categories of vector spaces (as in Proposition [efr-A6YR]). Note that we do not expect this type of example to present a Markov fibration. The reason is simply that, given a parametrized linear map M \times \mathbb {R}^n \to \mathbb {R}^n and a measure on M, the fact that the expectation map \mathbb {R}^n \to \mathbb {R}^n is the identity by no means implies that the original map is almost surely the identity or anything like that. If (m,e_0) \mapsto e_1 and m has positive probability, this can be canceled out by (m',e_0) \mapsto -e_1. This is impossible for probability kernels.

If we add an assumption of positivity, it seems plausible that examples of this type will present Markov fibrations---but of course, that brings us quite close to categories of Markov kernels in any case.