Notions of prechart equivalence [efr-3NFX]

Definition Behavioural equivalence [efr-X2S5]

Let \mathcal {D} be a monoidal stochastic module. Two precharts f_1, f_2: \bar {X} \to \bar {Y} with the same underlying map X \to Y are said to be behaviourally equivalent if, for any other three charts h: \bar {A} \to \bar {B}, s: \bar {S} \to \bar {A} \otimes \bar {X}, t: \bar {B} \otimes \bar {Y} \to \bar {T} so that the composite map S \to T is deterministic, the composite charts t(h \otimes f_1)s = t(h \otimes f_2)s agree as charts.

Lemma [efr-O92R]
  1. Sliding equivalence implies behavioural equivalence
  2. For charts over a deterministic base, sliding equivalence and behavioural equivalence coincide
  3. Given a faithful Markov functor \mathcal {C} \to \mathcal {C}' and a faithful map of modules \mathcal {D} \to \mathcal {D}' over it, the induced map on precharts in well-defined with respect to sliding equivalence, and injective with respect to behavioural equivalence, in the sense that precharts which become behavioural equivalent in the image are already behavioural equivalent, but not necessarily the other way around

The defect of behavioural equivalence is that \mathcal {D} may not have enough test charts to distinguish things which are "really" supposed to be different, while the defect of sliding equivalence is that there may not be enough maps to prove that two objects which are supposed to be equivalent, are.

Definition Absolute equivalence of charts [efr-CTIK]

Let \mathcal {D} be a stochastic module over \mathcal {C}, and let \phi ,\phi ' be parallel morphisms in \mathsf {SChart}(\mathcal {D}). We say \phi ,\phi ' are absolutely equivalent if there exists a faithful strong Markov functor \mathcal {C} \hookrightarrow \mathcal {C}', and a faithful map of stochastic modules \mathcal {D} \hookrightarrow \mathcal {D}' (with \mathcal {D}' over \mathcal {C}) so that \phi ,\phi ' are identified by the induced map \mathsf {SChart}(\mathcal {D}) \to \mathsf {SChart}(\mathcal {D}')

Note that by the above, sliding equivalence implies absolute equivalence implies behavioural equivalence.

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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