Categorical probability theory: Fritz' diagram categories

Question

1. Context.
In the seventh chapter of his 2019 paper A synthetic approach to categorical probability theory Tobias Fritz introduces certain diagram categories. Supposedly, these allow for a categorical approach to stochastic processes. Fritz illustrates this with an example — which I do not understand.

Consider the posetal category $\mathcal{D}=\mathbb{Z}$, where the objects are the integers and there is a morphism $n\rightarrow m$ if and only if $n\geq m$. Let $\mathcal{C}=\text{BorelStoch}$ be the category whose objects are standard Borel spaces and whose morphisms are Markov kernels between them. Note that the category $\mathcal{C}$ is a full subcategory of the Markov category $\text{Stoch}$ discussed in Chapter four of Fritz' paper. Next, consider the subcategory $\mathcal{C}_{\text{det}}$ of $\mathcal{C}$ which has the same objects as $\mathcal{C}$ but whose morphisms are the deterministic morphisms. In the case $\mathcal{C}=\text{BorelStoch}$, the deterministic morphisms can be identified with measurable maps between standard Borel spaces. Denote by $I$ the constant functor $\mathcal{D}\rightarrow \mathcal{C}_{\text{det}}$ that sends every integer to the singleton measurable space.

In chapter seven Fritz introduces the full subcategory $\operatorname{Fun}(\mathcal{D},\mathcal{C})$ of the functor category $[\mathcal{D},\mathcal{C}]$. Its objects are functors $\mathcal{D}\rightarrow \mathcal{C}_{\text{det}}$. Fritz indicates a correspondence between morphisms $I\rightarrow X$ in this category $\operatorname{Fun}(\mathcal{D},\mathcal{C})$ (i.e. natural transformations) and stochastic processes with indexing set $\mathbb{Z}$. He writes:

The functors $\mathcal{D}\rightarrow \mathcal{C}_{\text{det}}$ can be identified with infinite diagrams of measurable spaces $$ \ldots \rightarrow X_{n+1}\rightarrow X_n \rightarrow X_{n-1}\rightarrow \ldots$$ where all the maps involved are measurable functions. We think of each component $X_n$ as the space of joint values of a stochastic process over all times $t\leq n$, where the map $X_{n+1}\rightarrow X_n$ amounts to forgetting the value of the process at time $n+1$. A morphism $I\rightarrow X$ in $\operatorname{Fun}(\mathcal{D},\mathcal{C})$ then assigns a probability measure to each of the measurable spaces $X_n$ in such a way that the projections $X_{n+1}\rightarrow X_n$ are measure-preserving; thus such a morphism exactly turns $X$ into a stochastic process.

2. Questions.

• How can a morphism $I\rightarrow X$ in the functor category $\operatorname{Fun}(\mathcal{D},\mathcal{C})$ be seen as a (discrete-time) stochastic process? Fritz spells out what the objects and morphisms in the functor category $\operatorname{Fun}(\mathcal{D},\mathcal{C})$ are in our case. This I understand. However, I fail to see how to relate these structures to stochastic processes. This might be related to the Daniell-Kolmogorov extension theorem — I don't see how, though.
• Conversely, how does a (discrete-time) stochastic process give rise to a morphism $I\rightarrow X$ in the functor category $\operatorname{Fun}(\mathcal{D},\mathcal{C})$? My attempt at an explanation is given below.
• (In what sense) are these constructions inverses?
• What is meant by the following sentence?

We think of each component $X_n$ as the space of joint values of a stochastic process over all times $t\leq n$, where the map $X_{n+1}\rightarrow X_n$ amounts to forgetting the value of the process at time $n+1$.

3. Ideas
Let $(Y_i)_{i\in \mathbb{Z}}$ be a stochastic process with sample space $(\Omega, \mathcal{F}, P)$ and state space $(S,\Sigma)$. For each $n\in \mathbb{Z}$, define the set $X_n\colon =\{(Y_i(\omega))_{i\leq n}\ \vert \ \omega \in \Omega\}\subset \prod_{i\leq n}S$ and the projection $\pi_n\colon X_n\rightarrow X_{n-1}$ which deletes the $n$-th component. For each $n\in \mathbb{Z}$, consider the image $S_n\colon =\operatorname{im}(Y_n)$. For each $n\in \mathbb{Z}$, define the subspace $\sigma$-algebra $\Sigma_n\colon =\{A\cap S_n \ \vert \ A\in \Sigma\}$. Next, consider a pushforward measure on $(S_n,\Sigma_n)$, namely $\mu_n(A) \colon = P(\pi_n^{-1}(A))$ for $A \in \Sigma_n$. For each $n\in \mathbb{Z}$, we obtain a probability space $(S_n,\Sigma_n,\mu_n)$. Next, we define a $\sigma$-algebra $\mathcal{B}$ on the infinite product $\prod_{i\leq n}S_i$ as done here. Then we endow the pair $(\prod_{i\leq n}S_i, \mathcal{B})$ with the probability measure $\prod_{i\leq n}\mu_i$ as (again) defined here. Am I on the right track?

Answer 1

A previous version of the question implied that $\operatorname{Fun}(\mathsf{D},\mathsf{C})$ was notation for the functor category $[\mathsf{D},\mathsf{C}_\text{det}]$. This answer exists mostly to correct this misunderstanding, which I thought was the OP's main stumbling block. From the OP's feedback that seems not to be the case, so I might write another answer later answering their points in detail.

The important text is above propostion 7.1 (empahsis added)

Let $\mathsf{C}$ be a Markov category and $\mathsf{D}$ an arbitrary small category. We then introduce another Markov category $\operatorname{Fun}(\mathsf{D}, \mathsf{C})$. Its objects are defined to be functors $\mathsf{D} \to \mathsf{C}_\text{det}$, which we think of as diagrams with shape $\mathsf{D}$ of deterministic morphisms in $\mathsf{C}$, and its morphisms are natural transformations between these functors (with not necessarily deterministic components). We use notation $X = (X_d)_{d\in\mathsf{D}}$ for such functors, emphasizing their interpretation as diagrams.

This text is defining the category $\operatorname{Fun}(\mathsf{D},\mathsf{C})$ - it's not just notation for the functor category $[\mathsf{D},\mathsf{C}]$ or $[\mathsf{D},\mathsf{C}_\text{det}]$.

The objects of $\operatorname{Fun}(\mathsf{D},\mathsf{C})$ are functors $\mathsf{D} \to \mathsf{C}_\text{det}$, which is why the maps in the diagram

$$ \ldots \rightarrow X_{n+1}\rightarrow X_n \rightarrow X_{n-1}\rightarrow \ldots$$

are measurable functions and not Markov kernels. But the morphisms in $\operatorname{Fun}(\mathsf{D},\mathsf{C})$ are natural transformations in $\mathsf{C}$, not $\mathsf{C}_\text{det}$, so a morphism $I\to X$ is a family of Markov kernels (not measurable functions!) $I\to X_i$ for $i\in \mathbb{Z}$.