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Proposition [efr-2M0M]

May 6, 2025
Eigil Fjeldgren Rischel

There is a pullback diagram

in

\mathsf {DblCat}

Context

The triple category of dynamical systems [efr-MWCE]

May 6, 2025
Eigil Fjeldgren Rischel

Having done the prep-work in § [efr-000D] (and § [efr-0023]), we can jump directly into the construction:

Definition $\mathsf {BiSys}$ [efr-EPI9]

May 6, 2025
Eigil Fjeldgren Rischel

Let $\mathcal {A} \to \mathcal {C}, T: \mathcal {C} \to \mathcal {A}$ be a symmetric monoidal dynamical systems theory. Then this diagram:

depicts two strict double categories, each with a symmetric monoidal structure, and a strict double functor between them which is (non-strictly) a symmetric monoidal functor. This induces an object of

\mathsf {SymMon}(\mathsf {Act}(\mathsf {DblCat}))

. Applying

\mathsf {\mathbb Para}(-)

under

\mathsf {SymMon}(-),

we obtain a symmetric pseudomonoid in internal pseudocategories in

\mathsf {DblCat}

. Denote by

\mathsf {BiSys}(\mathcal {C},\mathcal {A},T)

this induced object.

Remark The Structure of $\mathsf {BiSys}$ [efr-WX1V]

May 6, 2025
Eigil Fjeldgren Rischel

$\mathsf {BiSys}(\mathcal {C},\mathcal {A},T)$ has the following structure: The objects are simply the objects of $\mathcal {A}$ ---the bundles.

There are three types of 1-cell: Lenses, that is morphism in $\mathcal {A}^\mathrm {fop}$ , which we write $A \leftrightarrows B$ , charts, that is morphisms in $\mathcal {A}$ , which we write $A \rightrightarrows B$ , and bisystems, which are pairs $(S \in \mathcal {C}, TS \otimes A \leftrightarrows B)$ , and which we write $A \nrightarrow B$ .

Moreover there are three types of 2-cell:

Lens-chart cells, which are the 2-cells of $\mathsf {\mathbb Arena}(\mathcal {A})$ . Note that this double category is thin. We will say a square of lenses and charts commutes if it is filled by such a 2-cell.
Chart-bisystem cells---given charts $A_1 \rightrightarrows B_1, A_2 \rightrightarrows B_2$ and systems $TS \otimes A_1 \leftrightarrows A_2, TS' \otimes A_2 \leftrightarrows B_2,$ a square filling this is a choice of map $S \to S' \in \mathcal {C}$ so that the resulting lens-chart square
commutes.
Lens-bisystem cells. Given lenses $A_1 \leftrightarrows B_1, A_2 \leftrightarrows B_2,$ and systems $TS' \otimes A_1 \leftrightarrows A_2, TS' \otimes B_1 \leftrightarrows B_2,$ a 2-cell consists of an isomorphism $S' \xrightarrow {\sim } S'$ so that the resulting square of lenses commutes (recall that isomorphism charts are the same as isomorphism lenses).

The charts, bisystems and chart-bisystem cells form a pseudo double category with the obvious composition. So do the lenses, bisystems and lens-bisystem cells. Finally, there is a notion of 3-cell given by a box whose sides are 2-cells of each kind, so that the resulting diagram in $\mathcal {C}$ (with isomorphisms on two sides) commutes.

The lens-bisystem double category is $\mathsf {\mathbb Para}_{\mathcal {C}^\simeq }(\mathsf {\mathbb Arena}{\mathcal {A}}_0)$ (recall that $\mathsf {\mathbb Arena}{\mathcal {A}}_0 = \mathcal {A}^\mathrm {fop}$ ), with the action given by the functor $T$ (restricted to isomorphisms). The chart-bisystem double category is the result of taking $\mathsf {BiSys}(\mathcal {C},\mathcal {A},T)$ , a pseudocategory in double categories, and applying $\mathsf {PsCat}((-)_h) : \mathsf {PsCat}(\mathsf {DblCat}) \to \mathsf {PsCat}(\mathsf {Cat})$ where $(-)_h$ takes the horizontal category of a (strict) double category.

The 3-cells, of course, are the 2-cells of $\mathsf {\mathbb Para}_{(\mathcal {C}^\to )^\simeq }(\mathsf {\mathbb Arena}(\mathcal {A})_1)$ . Analogously to the above, for each class of 1-cells, there is a double category with those as the objects, the two types of cell as the two morphisms, and the 3-cells as the 2-cells.

All of these six double categories admit a symmetric monoidal structure.

We can recover the ordinary category of (Moore) systems as the slice over $I$ , in the following sense:

Proposition [efr-2M0M]

May 6, 2025
Eigil Fjeldgren Rischel

There is a pullback diagram

in

\mathsf {DblCat}

In a double category $\mathbb {C}$ , there is a "horizontal slice" 1-category having objects the horizontal maps $A \to B$ and morphisms given by 2-cells that are identity on the left boundary, composed vertically. Similarly there is a "vertical slice". This is given by a similar pullback in $\mathsf {Cat}$ ---we simply do this one level up.

Note that Myers' construction of the double fibration $\mathsf {Sys}(\mathcal {C},\mathcal {A},T) \to \mathsf {\mathbb Arena}{\mathsf {Cat}}$ in fact uses the vertical slice in this sense.

This somewhat trivial observation means that any composition in $\mathsf {BiSys}(\mathcal {C},\mathcal {A},T)$ which produces 2-cell under $I$ in fact produces a morphism of systems in the ordinary sense. Replacing $I$ with another object, we may regard the slices as further-parametrized versions of $\mathsf {Sys}(\mathcal {C},\mathcal {A},T)$ .

There is essentially no difficulty in applying this to the Markov case:

Definition $\mathsf {BiSys}^M$ [efr-W413]

June 20, 2025
Eigil Fjeldgren Rischel

Let $(\mathcal {C}, \mathcal {A}, T)$ be a stochastic dynamical systems theory. Consider the double category whose vertical category is $\mathcal {C}_\mathrm {det}$ and horizontal category is $\mathcal {C}$ , with commutative squares as the 2-cells (this is the transpose of $\mathcal {C}^\to \rightrightarrows \mathcal {C}$ ). This carries an obvious symmetric monoidal structure, and acts on the double category of stochastic arenas $\widetilde {\mathsf {\mathbb Arena}}(\mathcal {A})$ via the functor $T$ . Denote by $\mathsf {BiSys}^M(\mathcal {C},\mathcal {A},T)$ the symmetric monoidal triple category induced as in Definition [efr-EPI9] by this data.

The information contained in $\mathsf {BiSys}^M(\mathcal {C},\mathcal {A},T)$ is much as above, although the complications involved in the double category of stochastic arenas remain present.

Our bisystems are reminiscent of the energy-driven systems of Capucci, Lynch, and Spivak (Reference [energy-driven-systems]). Indeed, their $\mathbb {C} \mathsf {org}$ is essentially the bisystems in the (ordinary) doctrine of smooth dynamical systems. As we mentioned in the introduction, Shapiro and Spivak (Reference [shapiro-spivak-dynamic-operads]) have previous developed a structure $\mathbb {O} \mathsf {rg}$ which consists of the bisystems for the discrete dynamical systems doctrine (i.e $\mathsf {Set}^\to \to \mathsf {Set}$ ). $\mathbb {O} \mathsf {rg}$ has a tremendous amount of structure coming from the representation of lenses in this doctrine as the category of polynomial functors, which can't be replicated for a general systems theory (and certainly not for a general stochastic systems theory).