Manuel observed that the two kinds of subobject of discrete possibilitic dynamical systems (dpds, )
have some semantic content directly relevant to boundaries. To wit, a subsystem *world* can never cross. Could I think about this as if, if I'm *own* boundary into parts of the world outside of me, and vice versa if I'm

OK, let's consider

I guess one case you could consider: two lax subobjects

A *discrete possibilistic dynamical system* (dpds) is a Kleisli morphism for the nonempty powerset monad *nonempty* powerset monad to be well behaved we may need to be in a *Boolean* topos. I'm not sure how necessary of a restriction this is. For possibilistic systems, think of something like one agent's model of the behavior of another agent with a comparable complexity. Especially if I don't know you personally, there's a range of things I imagine you might do while reading this paragraph: get bored and give up, go make a cup of tea, look at an email on your phone, or keep reading, and while in some sense I could probably assign probabilities to these outcomes, I think it is actually bad modeling to insist on pretending I'm doing so, when that's just not actually how I think about you: there are plausible actions, as well as things I'm substantially sure you won't do, and that's it.

Strict morphisms of dpds are the obvious thing: a map *a subsystem of a dpds, in the most obvious sense, is a subobject of the state space from which it is impossible to ever escape.*

This question of subsystems arose at the Boundaries workshop in the context of Nathaniel Osgood, Manuel Baltieri, Martin Biehl, and Matteo Capucci's work on a control theoretic approach to agent-environment boundaries. Here there is a specified "good" subspace of state space, which might be the subspace where the agent stays alive, and we look for a subobject of the system contained in this good space--this means a region of state space within which, having once entered, the agent will always remain. In generalizing from deterministic dynamical systems to possibilistic ones, then, the most natural move produces the notion of a set of states from which the agent can never depart, no matter what decision it makes. This is a natural and useful concept, but it's certainly not the only one we might propose. If we think the agent is smart and makes good decisions (hopefully), then it's enough for survival if the agent *can* always stay in the viable region.

To this end, consider a less obvious category whose objects are still dpds. For this, it's critical that the algebras for *lax* morphism of dpds using the square below; since powersets are just posets, there is a unique choice of the arrow filling the square, and all it says in elements is that

This condition on *possible*, rather than *unavoidable*, to remain permanently. Note that we've mildly modified the intuitive notion in that *extremal* monomorphisms of the category of dpds and lax maps. You might also note that this category of dpds and lax maps looks like it should be a 2-category; but I don't think there's any reasonable way to add 2-morphisms (which would be modifications) since the state spaces are discrete.