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Stream: theory: algebraic topology

Topic: Bisimulations of spaces

Jade Master (Mar 16 2021 at 17:16):

Following a discussion of bisimilarity in the #theory: concurrency stream a question occurred to me. Between two concurrent systems X and Y a bisimulation is roughly a third system A equipped with a covering of X and a covering of Y. The existence of these coverings says that X and Y have the same behavior. If two spaces X and Y have a common cover A, what does this say about the relationship between X and Y? Is there some relation between their homotopy types? what sort of properties do they share?

Fawzi Hreiki (Mar 16 2021 at 17:36):

Assuming both spaces are nonempty, isn’t the sum of the two spaces always a covering of both?

Fawzi Hreiki (Mar 16 2021 at 17:37):

You just map one side to the space itself and the other side constantly to any point. This gives a split epimorphism since its a retraction of the injection map.

John Baez (Mar 16 2021 at 17:40):

Jade Master said:

If two spaces X and Y have a common cover A, what does this say about the relationship between X and Y? Is there some relation between their homotopy types? what sort of properties do they share?

A space has a universal cover if it's reasonably nice, like a manifold or the space coming from simplicial set. Let's not get into this; I think your question has good answer for reasonably nice spaces and becomes a bit scary otherwise.

Henceforth all my spaces will be "reasonably nice".

Two spaces have a common cover iff they have the same universal cover.

So, for reasonably nice spaces your question becomes "if two spaces have the same universal cover, what does this say about their relation?"

People often phrase it as "what does a space have in common with its universal cover?"

And here there is a pretty nice bunch of answers. Obviously, each point $x$ in the space is covered by a bunch of points in the universal cover, and a neighborhood of $x$ looks just like a neighborhood of each these points. So we say "locally, the universal cover looks just like the original space".

So, the fun questions are global, not local. We should look at topological invariants.

For starters, a space has the same homotopy groups $\pi_n$ as its universal cover except for the fundamental group, $\pi_1$. The universal cover has $\pi_1 = 0$. The whole point of taking the universal cover is to kill off $\pi_1$ while doing as little damage to a space as possible.

I don't know much about the homology groups of the universal cover - a hole in my education! $H_1$ of any space is the abelianization of $\pi_1$ of that space, which is $0$ for the universal cover, so $H_1 = 0$ for the universal cover.

$H_2 \cong \pi_2$ for any space that has $\pi_1 = 0$, by the Hurewicz theorem, so $H_2$ of the universal cover is the same as $\pi_2$ of the universal cover, which is the same as $\pi_2$ of the original space.

Around here my knowledge fizzles out.

Fawzi Hreiki (Mar 16 2021 at 17:41):

Scratch what I said. That’s definitely not a covering.

Jade Master (Mar 16 2021 at 18:27):

Interesting @John Baez the universal cover idea also makes sense for Petri nets. As @Joachim Kock explained the universal cover of a Petri net is it's unfolding...and certainly two Petri nets have the same behavior if they gave the same unfolding.

John Baez (Mar 16 2021 at 18:52):

I don't understand the concept of "unfolding" for a Petri net - at least, I've never learned what it is. But I can imagine the universal cover of a Petri net. It seems a lot like the universal cover of a graph. And the universal cover of a graph is a lot simpler than the universal cover of a general space - since for graphs the only nonzero homotopy group is $\pi_1$, so killing $\pi_1$ is "killing everything".

To put it more precisely: the universal cover of any connected graph is contractible. So, for homotopy theorists it's an utterly boring space, homotopy equivalent to a point.

It can still be interesting in other ways because it's not homeomorphic to a point. Here's the universal cover of the space shaped like the number 8:

universal cover of the figure 8

Since $\pi_1$ of the figure 8 is the free group on 2 generators, here we have a bunch of points corresponding to all the elements of the free group on 2 generators $a$ and $b$.

John Baez (Mar 16 2021 at 18:53):

I can imagine doing something like this to a Petri net.

Frederick Kintanar (Mar 16 2021 at 19:16):

I am trying to imagine a practical problem domain for this situation. If X and Y are the neurocognitive systems of two different individuals, each might be covered by the same scheme of lexicalized concepts A, and by the algebra generated via a shared grammar system. The neural systems X and Y are "the same" but not on the nose physically. Their equivalence is only up to the identifying conditions of the scheme of lexicalized concepts, a scheme that is acquired by social interaction. So concept-like structures are a coarse-grained classification of the world of experience, with different realizations in neural correlates. The neural correlates of individuals do not need to the same in a fine-grained physicalist sense, as long as they support the same coarse-grained schemes up to equivalence in the specified context of a shared linguistic system.

Classifying the world(s) of experience in a precise manner is a big challenge, a hard problem. Can homotopy types or cohomology tell us something about concept types and their realizations? Or someday provide a basis from mathematical models of language and cognition at this level?

John Baez (Mar 16 2021 at 19:50):

Frederick wrote:

Can homotopy types or cohomology tell us something about concept types and their realizations? Or someday provide a basis from mathematical models of language and cognition at this level?

Oof! Someday, maybe... but not today, unless someone out there is doing work in secret. :upside_down:

John Baez (Mar 16 2021 at 19:53):

It's good to watch this:

Kathryn Hess on categorical approaches to neuroscience

If I recally correctly it mainly focuses on homotopy theory (she is a homotopy theorist), and the overall conclusion is: apart from using persistent homology to study neuroscience data, applications of homotopy theory or category theory to neuroscience are mainly just hopes at this point.

(This talk was in 2018.)

Joachim Kock (Mar 16 2021 at 19:53):

I think the analogy between unfolding and universal cover is good enough as an analogy, but the notion of simply-connectedness should probably be analysed a bit more, because it is surely more subtle than the graph-theoretic notion. As far as I understand the condition characterising unfolding it is: directed hypergraph such that every hyper-edge is outgoing of at most one node (and one should think node=transition. There are two conditions here that hint in direction of simply-connectedness: one is that it is a hypergraph rather than a general Petri net, meaning that there are no parallel arcs. And the second condition which is to ensure that hyperedges have unique history when interpreted as token occurrences. I don't konw to what extend these two conditions really express any simply-connectedness... As John points out, it would be boring if all Petri nets were bisimilar!

Frederick Kintanar (Mar 16 2021 at 19:59):

John Baez said:

It's good to watch this:
[Kathryn Hess on categorical approaches to neuroscience]

Thanks for the link! I also liked Toby St. Clere Smithe's talk (I think in the MIT category theory seminar) about neuroscience models as cybernetic systems, via categories and probability theory.

Morgan Rogers (he/him) (Mar 16 2021 at 20:56):

The universal cover should work for directed graphs too (or even graphs with labelled edges), and that's how I think of Petri nets in my limited experience of them. You just decorate the edges into or out of a given place to match the net that you're covering, right?
This gives more variety than the universal covers of the underlying graphs, though, because you have to preserve what the decorations look like locally as well as the topological stuff.

Jade Master (Mar 17 2021 at 14:34):

Joachim Kock said:

I think the analogy between unfolding and universal cover is good enough as an analogy, but the notion of simply-connectedness should probably be analysed a bit more, because it is surely more subtle than the graph-theoretic notion. As far as I understand the condition characterising unfolding it is: directed hypergraph such that every hyper-edge is outgoing of at most one node (and one should think node=transition. There are two conditions here that hint in direction of simply-connectedness: one is that it is a hypergraph rather than a general Petri net, meaning that there are no parallel arcs. And the second condition which is to ensure that hyperedges have unique history when interpreted as token occurrences. I don't konw to what extend these two conditions really express any simply-connectedness... As John points out, it would be boring if all Petri nets were bisimilar!

Thanks that's a good point. The point of unfolding is also to get rid of all the cycles...I'm imagining it more like a tree but as a hypergraph rather than a graph. I'll have to keep thinking about this.

Jade Master (Mar 17 2021 at 14:38):

Morgan Rogers (he/him) said:

The universal cover should work for directed graphs too (or even graphs with labelled edges), and that's how I think of Petri nets in my limited experience of them. You just decorate the edges into or out of a given place to match the net that you're covering, right?
This gives more variety than the universal covers of the underlying graphs, though, because you have to preserve what the decorations look like locally as well as the topological stuff.

Maybe...this is all something which I've gotta think about a bit more.

Joachim Kock (Mar 17 2021 at 14:47):

Meanwhile I am getting more and more confused about the analogy between universal unfolding and universal cover. One is a terminal object, the other is an initial object, as John explained. If someone figures it out it would be great. (There is also the possibility that what I call unfolding is not the same as what WInskel calls unfolding --- after all the notions of Petri net are not the same...)

Fabrizio Genovese (Mar 17 2021 at 16:09):

Actually, a useful thing to do in practice is "unfolding up to n". As soon as your net has one cycle, your unfolding net will be infinite. This causes all sort of practical problems because an infinite net is difficult (if not impossible) to store. A practical solution is to unfold the net in a way parametrized by an index. For instance, if you have a cycle, you can unfold it 3 times. The resulting net only captures a part of the possible executions of the net you started from, obviously, but if your starting net is finite, it will be finite as well, and usually much simpler reachability-wise.

Fabrizio Genovese (Mar 17 2021 at 16:10):

I don't know how this translates in terms of covers, but my point is that the "obvious" universal object may not be the only interesting thing here.

Mike Shulman (Mar 17 2021 at 16:19):

Fabrizio Genovese said:

an infinite net is difficult (if not impossible) to store.

Couldn't it be a coinductive object?

Fabrizio Genovese (Mar 17 2021 at 16:22):

Yes, it could. :smile: