There is one more blog to mention: John Baez has recently been writing about PROPs and linear systems; check it out if you want a sneak preview of one of the star applications of graphical linear algebra! John and his student Jason Erbele developed the theory independently and more or less at the same time as Filippo, Fabio and I. But, apart from the fact that they have different conventions for drawing diagrams, the equational theory that they have developed can be shown to be equivalent to the one we will develop here. It’s super interesting that science is full of temporal coincidences like this; it’s not even the first time that this kind of thing happens to me!

]]>If our field is the good old real numbers, we write as an abbreviation for The direct sum of objects is

which we abbreviate as addition of natural numbers: adding and gives We write

to mean a linear map

or more concretely an matrix. We compose these maps by matrix multiplication. And given linear maps

and

we get a linear map

corresponding to the block diagonal matrix with and as its blocks.

All this stuff is called the PROP Then you can replace by any other commutative rig.

]]>[…] composition of morphisms is given by matrix multiplication, and the tensor product of morphisms is the direct sum of matrices.

At first I read this as saying that the tensor product of morphisms was given by ordinary element-by-element matrix addition, which seemed remarkably weird. Then I realized, “Oh, right, *direct* sum.”

If you don’t mind, I will use your comments to plug my blog:

http://graphicallinearalgebra.net/

where I plan to develop all this, but taking a much more leisurely pace. I have just written about bicommutative bimonoids over the weekend, and I plan to spend some time discussing the various approaches to proving the results that you mention here. It seems that the Wadsley Woods papers gives a really nice new perspective on things!

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