Yes, we’re getting old. The main problem with that (for me, right now) is that I feel unmotivated to re-explain things that I explained with great excitement 10 or 20 years ago. I think some people may wonder why I’m working on electrical circuits now. If I had the energy, I’d start with an explanation of why n-categories are so great, how they’re going to revolutionize our understanding of math and physics, and why we need to understand complex systems… and *then* I’d start talking about how to use n-categories (for n = 1 and 2) to think about electrical engineering in new ways. But I’m tired of giving the sales pitch—I want to just dive in and do the real work.

If you ever want to talk more about quantropy, let me know. Blake and I haven’t gotten around to publishing our paper on that, and while I think they’re some really exciting things left to understand, he’s moved on to other projects.

]]>You just tell me the properties it has and the operations I am allowed to perform. How it is constructed does not interest me the slightest ;-) Like with real numbers.

That’s an interesting example. In the category of sets the construction of the reals using Cauchy sequences gives a result isomorphic to the construction using Dedekind cuts. But in other topoi (i.e., other categories very much like the category of sets) the Cauchy reals and the Dedekind reals can differ!

I’ve never had any need to work with real numbers in a category other than the category of sets, so this seems rather abstruse to me. However, experts tell me that the Dedekind reals behave better.

• Cauchy real number, nLab.

On the other hand, I’ve been interested in graphs in many different categories, so this is a live issue to me.

]]>They were always **very inspiring** and **motivating** and I very much hope you **keep them coming**.

The quantropy series I liked a lot and did indeed some work on it. During that time I convinced myself that I need some serious knowledge of quantum field theory to progress further. That was beyond my resources and I shelved the project. I still think (maybe wishfull thinking) that with the ideas you have developed one can probably determine a minimal length scale for concrete quantum systems.

I think that your quantropy series was a prequel to what later became polymath projects. Your intention was different, but at least that was how I perceived it.

]]>One reason it pays to analyze how a structure is built is that this gives clues on when we’ll be able to build analogous structures in other contexts.

From what I said, the concept of graph that I’m using in this course will generalize from the category of sets to any category whatsoever. The kind of graph you’re talking about will generalize to any category with binary products and reasonably well-behaved subobjects.

You see, in any category we can talk about

but it takes more bells and whistles to talk about

Here we are talking about subobjects () and binary products (). There are still tons of categories that have nicely behaved subobjects and binary products, so that’s still a huge class of contexts, but I like to leave my options open as long as possible. Of course if there were some reason I *needed* to use graphs of the sort I would do it in an instant.

For the categorically minded:

**Puzzle.** Using the former concept of graph, what’s a graph in the category of vector spaces? More precisely, what well-known category is equivalent to the category of graphs in the category of vector spaces?

How mathematicians have different perspectives on the same object! For me is (or maybe was) an abstract data type. You just tell me the properties it has and the operations I am allowed to perform. How it is constructed does not interest me the slightest ;-) Like with real numbers.

]]>Thévenin could be mentioned here.

]]>Anyway, I sense a strong family resemblance between simplicial complexes and simple graphs: I guess the latter can be seen as the 1-dimensional ‘truncation’ of the former. (Somehow the product of two simple graphs is still 1-dimensional in this truncation, while in the category of simplicial sets it would be 2-dimensional. But this reminds me of other things that work this way, like the product of categories.)

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