Up to light editing, the following was written by me during the “Finding the Right Abstractions for healthy systems” research workshop, hosted by Topos Institute in January 2023. However, I invented the idea before.
In order to allow R (the set of programs) to be infinite in IBP, we need to define the bridge transform for infinite Γ. At first, it might seem Γ can be allowed to be any compact Polish space, and the bridge transform should only depend on the topology on Γ, but that runs into problems. Instead, the right structure on Γ for defining the bridge transform seems to be that of a “profinite field space”: a category I came up with that I haven’t seen in the literature so far.
The category PFS of profinite field spaces is defined as follows. An object F of PFS is a set ind(F) and a family of finite sets Fαα∈ind(F). We denote Tot(F):=∏αFα. Given F and G objects of PFS, a morphism from F to G is a mapping f:Tot(F)→Tot(G) such that there exists R⊆ind(F)×ind(G) with the following properties:
For any α∈ind(F), the set R(α):=β∈ind(G)∣(α,β)∈R is finite.
For any β∈ind(G), the set R−1(β):=α∈ind(F)∣(α,β)∈R is finite.
For any β∈ind(G), there exists a mapping fβ:∏α∈R−1(β)Fα→Gβ s.t. for any x∈Tot(F), f(x)β:=fβ(prRβ(x)) where prRβ:Tot(F)→∏α∈R−1(β)Fα is the projection mapping.
The composition of PFS morphisms is just the composition of mappings.
It is easy to see that every PFS morphism is a continuous mapping in the product topology, but the converse is false. However, the converse is true for objects with finite ind (i.e. for such objects any mapping is a morphism). Hence, an object F in PFS can be thought of as Tot(F) equipped with additional structure that is stronger than the topology but weaker than the factorization into Fα.
The name “field space” is inspired by the following observation. Given F an object of PFS, there is a natural condition we can impose on a Borel probability distribution on Tot(F) which makes it a “Markov random field” (MRF). Specifically, μ∈ΔTot(F) is called an MRF if there is an undirected graph G whose vertices are ind(F) and in which every vertex is of finite degree, s.t.μ is an MRF on G in the obvious sense. The property of being an MRF is preserved under pushforwards w.r.t.PFS morphisms.
Up to light editing, the following was written by me during the “Finding the Right Abstractions for healthy systems” research workshop, hosted by Topos Institute in January 2023. However, I invented the idea before.
In order to allow R (the set of programs) to be infinite in IBP, we need to define the bridge transform for infinite Γ. At first, it might seem Γ can be allowed to be any compact Polish space, and the bridge transform should only depend on the topology on Γ, but that runs into problems. Instead, the right structure on Γ for defining the bridge transform seems to be that of a “profinite field space”: a category I came up with that I haven’t seen in the literature so far.
The category PFS of profinite field spaces is defined as follows. An object F of PFS is a set ind(F) and a family of finite sets Fαα∈ind(F). We denote Tot(F):=∏αFα. Given F and G objects of PFS, a morphism from F to G is a mapping f:Tot(F)→Tot(G) such that there exists R⊆ind(F)×ind(G) with the following properties:
For any α∈ind(F), the set R(α):=β∈ind(G)∣(α,β)∈R is finite.
For any β∈ind(G), the set R−1(β):=α∈ind(F)∣(α,β)∈R is finite.
For any β∈ind(G), there exists a mapping fβ:∏α∈R−1(β)Fα→Gβ s.t. for any x∈Tot(F), f(x)β:=fβ(prRβ(x)) where prRβ:Tot(F)→∏α∈R−1(β)Fα is the projection mapping.
The composition of PFS morphisms is just the composition of mappings.
It is easy to see that every PFS morphism is a continuous mapping in the product topology, but the converse is false. However, the converse is true for objects with finite ind (i.e. for such objects any mapping is a morphism). Hence, an object F in PFS can be thought of as Tot(F) equipped with additional structure that is stronger than the topology but weaker than the factorization into Fα.
The name “field space” is inspired by the following observation. Given F an object of PFS, there is a natural condition we can impose on a Borel probability distribution on Tot(F) which makes it a “Markov random field” (MRF). Specifically, μ∈ΔTot(F) is called an MRF if there is an undirected graph G whose vertices are ind(F) and in which every vertex is of finite degree, s.t.μ is an MRF on G in the obvious sense. The property of being an MRF is preserved under pushforwards w.r.t.PFS morphisms.