Deletion can be written as a composite using only duplication and cup. Similarly, the generator ‘zero’ can be written as a composite using addition, scalar multiplication by -1, and cap. That one could be considered cheating if you don’t want to take scalar multiplication by -1 to be fundamental, but if you Frankenstein stuff together, you might be able to get the ‘zero’ generator from cap, cup, duplication, and addition.

Notwithstanding comment 1, it is plain that the list of generators, where is the number of elements in the field , has some (extreme, in some cases) redundancy. None of the infinitely many scalar multiplication generators are necessary for , for instance. We can get them all “for free” because is the smallest field that contains the rig , which we get “for free” from duplication and addition of identity morphisms. Scalar multiplication generators only become necessary when you do something crazy like take to be the field . There you really do need scalar multiplication by (or some rational multiple thereof) to be in your set of generators.

Summing up comments 1 and 2, any linear transformation of vector spaces over (probably)* only *requires*:

.

It would not be nice to read or draw.

I can think of another way to make a list of only four generators that does the same job and definitely works, but I will leave that as a puzzle.

*I’m approximately 95% confident ‘zero’ doesn’t need to be on the list I give here.

]]>You noted that in the generator ‘zero’

is the composite of codeletion

and scalar multiplication by zero

It should be a fun puzzle to derive this from the relations we listed, just to check that Jason found enough relations!

But I also think the generator ‘deletion’

should be the composite of scalar multiplication by zero

and cozero

So in the end I don’t think zero is any more or less redundant than deletion. We’re more interested in a symmetrical and beautiful set of generators than a minimal set.

It takes a while to get used to coaddition, cozero, codeletion and coduplication, since they forgot to tell us about those in school.

]]>The zero generator looks a bit redundant, because you can write it as scalar multiplication by followed by codeletion. Can you similarly write addition in terms of the other generators? (I guess not, but how do you prove this?)

The diagrams used here remind me of linear network coding. There, one also likes to draw diagrams in , considered as a monoidal category under direct sum, where is some finite field.

]]>Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now it’s time for me to say what my students and I have doing.

]]>Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now it’s time for me to say what my students and I have doing.

]]>Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now it’s time for me to say what my students and I have doing.

]]>