Slice Markov Categories [efr-OKM1]

Definition Slice Markov Category [efr-EANM]

Let \mathcal {C} be a Markov category. Suppose \mathcal {C} admits deterministic pullbacks. Then for each object X \in \mathcal {C}, let \mathcal {C}_{/X} denote the category where objects are deterministic maps A \to X and morphisms are commutative triangles (whose map A \to B is not necessarily deterministic).

Note that \mathcal {C}_{/X} admits the structure of a Markov category---given A_1 \to B_1 and A_2 \to B_2 over X, the induced map A_1 \times _X A_2 \to B_1 \times B_2 necessarily lands within the subobject B_1 \times _X B_2

Lemma [efr-297V]

Let \mathcal {C} be a Markov category with good deterministic pullbacks, and let X \in \mathcal {C}

  1. If \mathcal {C} is causal, so is \mathcal {C}_{/X}.
  2. If P \to A \in \mathcal {C}_{/X} admits a support S \subseteq A in \mathcal {C}, then this is also a support in \mathcal {C}_{/X}

Proof

Since the inclusions A \times _X B \to A \times B are all monomorphisms, it is easy to verify causality. Similarly, it is easy to see that being a support in \mathcal {C} is simply stronger than being a support in \mathcal {C}_{/X}.

Backlinks

Projects [efr-VJB4]

Some current projects:

  1. My PhD thesis Markov Fibrations. I am working on a "journal version" of the main contribution, the eponymous theory of Markov fibrations/stochastic module fibrations. This will probably also be the home for some minor contributions to Markov category theory, especially the theory of Slice Markov Categories.
  2. Tangencies of commutative monoids. There should be a theory like this which interpolates between petri nets and DOTS-type dynamical systems (note that the state space of the dynamical system associated to a petri net is the free commutative monoid on the set of places). With Owen Lynch and David Jaz Myers.
  3. Control theory/lyapunov functions for DOTS. There is a nice way to cook up a category of "systems equipped with invariant subspaces"/systems equipped with Lyapunov functions as an almost free enhancement to the theory of systems. Combining this idea with the triple category of "bisystems" has the potential to produce a very general approach to control theory. With Matteo Capucci.
  4. Universal property of measure-theoretic probability theory. Nearly finished paper, The Universal Property of Measure-Theoretic Probability.
  5. Lack's discrete or trivial model structure on a generic 2-category is self-dual (in the sense that the dual model structure on \mathcal {C}^\mathrm {op} is exactly the one associated to the 2-category \mathcal {C}^\mathrm {op}), something which simplifies a number of things in Lack's paper.
  6. The tangency of martingale problems and representable dynamics for martingale problems. Almost done (written up on the safeguarded AI forest), should be made public soon.

Random crap:

  1. Markov Fibrations
  2. Lyapunov function
  3. Slice Markov Categories
  4. https://erischel.com/efr-ECP8/
  5. Notions of prechart equivalence