• Malin Christensen, Rolling hypocycloids and epicycloids, 18 November 2015.

It’s so old I’m surprised I hadn’t seen it before! Or maybe I had and I’ve forgotten. It was brought to my attention by Alok Tiwari on G+.

]]>(thank you for fixing my syntax!)

]]>By the way, to use TeX here, you have to enclose the math with

$latex $

with the word ‘latex’ appearing directly after the first dollar sign, no space between, but then a space after it.

]]>More precisely, when you compose two morphisms, some blocks may lose contact to the input and the output. In this case, these blocks (let’s say there are of them) are omitted and the result is multiplied with the -th power of a parameter, usually denoted .

When is small, there is a basis of the centralizer algebra indexed (!) by set partitions into at most blocks.

Still, there are some mysterious open questions…

]]>Davis’s law of small devices:

A device can be made arbitrarily small if it doesn’t have to work.

• John Baez and Brendan Fong, A compositional framework for

passive linear networks. (Blog article here.)

or my paper with Blake Pollard on reaction networks:

• John Baez and Blake Pollard, A compositional framework for reaction networks.

will find many of Darbo’s ideas eerily similar.

]]>• G. Darbo, Aspetti algebrico-categoriali della teoria dei dispotivi, *Symposia Mathematica* **IV** (1970), Istituto Nazionale di Alta Matematica, 303–336.

It’s closely connected to Brendan Fong’s thesis, but also different—and, of course, much older. According to Grandis, Darbo was the first person to work on category theory in Italy. He’s better known for other things, like ‘Darbo’s fixed point theorem’, but this piece of work is elegant, and, it seems to me, strangely ahead of its time.

The paper’s title translates as ‘Algebraic-categorical aspects of the theory of devices’, and its main concept is that of a ‘universe of devices’: *a collection of devices of some kind that can be hooked up using wires to form more devices of this kind*. Nowadays we might study this concept using operads—but operads didn’t exist in 1970, and Darbo did quite fine without them.

The key is the category which has finite sets as objects and ‘corelations’ as morphisms. I explained corelations here:

• Corelations in network theory, 2 February 2016.

]]>• G. Darbo, Aspetti algebrico-categoriali della teoria dei dispotivi, *Symposia Mathematica* **IV** (1970), Istituto Nazionale di Alta Matematica, 303–336.

It’s closely connected to Brendan Fong’s thesis, but also different—and, of course, much older. According to Grandis, Darbo was the first person to work on category theory in Italy. He’s better known for other things, like ‘Darbo’s fixed point theorem’, but this piece of work is elegant, and, it seems to me, strangely ahead of its time.

]]>A good illustration of this is Google’s TensorFlow project. It’s an “open-source software library for machine intelligence”, which makes it easy for people to write their own software using neural networks.

The word “TensorFlow” makes me think of monoidal categories and string diagrams—and I don’t think I’m being silly. Here is their summary:

TensorFlow™ is an open source software library for numerical computation using data flow graphs. Nodes in the graph represent mathematical operations, while the graph edges represent the multidimensional data arrays (tensors) communicated between them.

So there are definitely monoidal categories lurking in the background here! But since this is a highly popular area, I only want to get involved if I can do something interesting: these days I dislike bandwagons, so I only want to jump aboard one if I can take it in a slightly new direction.

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