• Eugene Lerman and David Spivak, An algebra of open continuous time dynamical systems and networks.

open electrical circuits and chemical reaction networks:

• Kenny Courser, A bicategory of decorated cospans, *Theory and Applications of Categories* **32** (2017), 995–1027.

open discrete-time Markov chains:

• Florence Clerc, Harrison Humphrey and P. Panangaden, Bicategories of Markov processes, in *Models, Algorithms, Logics and Tools*, Lecture Notes in Computer Science **10460**, Springer, Berlin, 2017, pp. 112–124.

and coarse-graining for open continuous-time Markov chains:

• John Baez and Kenny Courser, Coarse-graining open Markov processes. (Blog article here.)

As noted by Shulman, the easiest way to get a symmetric monoidal bicategory is often to first construct a symmetric monoidal double category:

• Mike Shulman, Constructing symmetric monoidal bicategories.

]]>Lemma 5.3 is about what happens when we apply the 2-functor , which is *black-boxing*, to 2-morphisms in Mark, which are *morphisms between open Markov processes*. The simplest morphisms between open Markov processes are *coarse-grainings*. So, the Lemma says what happens when we black-box a coarse-graining.

The simplest case is when the maps and in the diagram are identity functions. This means that we’re leaving the input and output states alone. In this case, Lemma 5.3 says that the relation between input and output probabilities and flows in steady state are *the same* for the original open Markov process and the coarse-grained open Markov process.

I thought that might be what you were trying to say here:

I could have imagined e.g. that an internal lumpability might for example not change the relations between input and outputs of a Markov network.

This is true if by “the relations between inputs and outputs” you mean “the relation between input and output probabilities and flows in steady state”. This relation is what black-boxing gives us.

The double category formalism, with its supposedly cryptic diagrams, is just a way of drawing open Markov processes and the morphisms between them, e.g. coarse-grainings. With this style of picture, the coarse-grainings look like rectangles. The basic operations on coarse-grainings look like ways of sticking together rectangles to get bigger rectangles. The rules these operations obey turn into intuitive rules about sticking together rectangles.

That’s the whole point of double categories. Double categories are “the algebra of rectangles”, just as categories are “the algebra of arrows”.

We don’t explain this, because this paper was written for people who are comfortable with double categories. Explaining double categories would double the length of this paper.

I said morphisms between open Markov processes look like rectangles. But the simplest morphisms between open Markov processes are coarse-grainings, so let me talk just about coarse-grainings. The two main operations on these are:

1) We can compose coarse-grainings of open Markov processes “horizontally” by attaching the outputs of one to the inputs of another.

2) We can compose coarse-grainings of open Markov processes “vertically” by doing first one and then another: coarse-graining and then coarse-graining some more.

We can stick together two rectangles either horizontally or vertically, and that’s how we draw operations 1) and 2). Some nontrivial laws governing these operations then drawn as obvious-looking laws about sticking rectangles together.

]]>It sounds like you’re talking about Lemma 5.3, which says that coarse-graining is compatible with black-boxing.

Well, as said I was looking for a remark like that, but I didn’t and unfortunately still don’t see that “coarse-graining is compatible with black-boxing” is explained in Lemma 5.3:

Lemma 5.3: Given a 2-morphism [DIAGRAM] in Mark, there exists a unique 2-morphism [DIAGRAM] in LinRel.

That is even if you tell me that “build the equation into the definition of ‘morphism between open Markov processes” I don’t see why Lemma 5.3. says “coarse-graining is compatible with black-boxing” and in particular what you mean with “compatible.” This seems to be hidden in those cryptic diagrams. But anyways I got now a rough idea of what you were doing here.

]]>I could have imagined e.g. that an internal lumpability might for example not change the relations between input and outputs of a Markov network, but I haven’t found a remark similar to that.

It sounds like you’re talking about Lemma 5.3, which says that coarse-graining is compatible with black-boxing. You’ll note that the proof uses the equation

In our paper we define ‘lumpability’ in Definition 3.4. Theorem 3.10 gives 3 equivalent conditions for lumpability. The best one is condition (ii):

This is clearly equivalent to

This condition says that two things are the same:

1) evolving a state in time using our original Markov process with Hamiltonian and then coarse-graining it with ,

2) coarse-graining a state with and then evolving it in time using the coarse-grained Markov process with Hamiltonian .

Since this is so nice, we build the equation into the definition of ‘morphism between open Markov processes’ in Definition 3.4. In the rest of the paper we study these morphisms and never mention the word ‘lump’.

We need the condition to prove most of the interesting results in the paper. For example, Lemma 5.3.

The literature on Markov processes talks about lumpability, but they usually use the equivalent condition (iii) as their definition of lumpability.

]]>thanks for replying. You wrote:

Later on, for category theoretic reasons, we decided to only work with Markov processes that were lumpable, since lumpability is precisely the condition needed for the Hamiltonian of the new Markov process to be independent of the choice of stochastic section s.

Frankly I have problems to see why this is a category theoretic question. In particular I haven’t sofar understood what this lumpability is needed for. I could have imagined e.g. that an internal lumpability might for example not change the relations between input and outputs of a Markov network, but I haven’t found a remark similar to that.

I searched the text for all occasions of the word “lump”, but apart from the reference it didn’t appear in the text after theorem 3.10.

]]>The image in the tweet was taken from back when we were originally considering more general coarse-grainings of Markov processes. Later on, for category theoretic reasons, we decided to only work with Markov processes that were lumpable, since lumpability is precisely the condition needed for the Hamiltonian of the new Markov process to be independent of the choice of stochastic section s. As you noticed, the image in the tweet isn’t lumpable (at least not in the manner that is being visually suggested) since the rates of the edges going from both of the states w1 and w2 to the state y are not the same. They are in the example in the paper.

It’s also perhaps worth noting that the edge with rate 4 going from the state w2 to the state w1 plays no role in the lumped Markov process, which intuitively makes sense: anything in either of the states w1 or w2 will transition to state y at the same rate in the case where the original Markov process is lumpable, and transitioning between states w1 and w2 will not affect this.

]]>OK thanks. Now I am more or less sure that I understand what you mean with the notation . By the way is Kenny reading the comments to this blog post?

As said I might have a different approach to derive a Hamiltonian from a diagram, but with the approach I sofar used, the Hamiltonian as derived with my approach would not be lumpable:

@Paul

I don’t see what this have to do with stiffness problems, but if 4 gets big then with the above hamiltonian the w to y transition would converge to 9 only for a specific section.

If the rate of 4 that describes the w2 to w1 transition was increased to a very large number, the coarse-grain solution would result in the w to y transition asymptotically converging to 9. This is a basic form of order-reduction for stiff problems.

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