# Categorical probability theory: Fritz' diagram categories

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?

• What is a "deterministic morphism"? Commented Jun 16, 2023 at 16:22
• @FShrike: A morphism $f\colon X \rightarrow Y$ in a Markov category is called deterministic if $\Delta_Y \circ f=(f\otimes f)\circ \Delta_X$ holds. Here, $\Delta_X$ and $\Delta_Y$ denote the respective copy maps/comultiplications in the Markov category. You can think of this as follows: Copying and the applying $f$ to each copy separately yields the same result as first applying $f$ and then copying the result. In more informal terms (due to Paolo Perrone): "Randomness is doing the same thing twice and expecting different results." (The term "expected" seems a bit unfortunate in this context.) Commented Jun 16, 2023 at 16:42

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 $$i\in \mathbb{Z}$$.

• Thanks for correcting my mistake. That still does not explain the connection with stochastic processes, though. Commented Jun 14, 2023 at 14:43
• @M.C. your idea in part 3 of your question is basically correct - I think if you look at the text again in light of this definition you should be able to make sense of it. Commented Jun 14, 2023 at 14:45
• I don't, I am afraid. How does a morphism $I\rightarrow X$ in $\operatorname{Fun}(\mathcal{D},\mathcal{C})$ give rise to a sequence of $\mathbb{Z}$-indexed random variables? Commented Jun 16, 2023 at 8:11