Alexander Ellis has just put up a very cool post over at Concrete Nonsense, called Wood, Glue, and the Octahedral Axiom. For those of you who don’t know, the Octahedral Axiom is the Fourth and Most Annoying Axiom for triangulated categories. It’s a nuisance for two reasons. First, it’s a bit aesthetically unsatisfactory. One of the triangulated category axioms asserts that for any map , there exists an exact triangle
The “cokernel” isn’t unique up to unique isomorphism — this is the aesthetically unsatisfying bit — so we’re forced to introduce the Octahedral Axiom, which asserts that , while not canonical, is at least somewhat functorial. Second, the Octahedral Axiom is usually written as (the projection onto the plane/page) of a certain partially commutative three-dimensional octahedron diagram. It’s sort of a pain to see what the various parts of this diagram mean.
Anyways, Ellis has gotten crafty and put together a miniature model of the Octahedral Axiom. (The real one is kept in Paris, in the basment of IHES, and is quite large, but Ellis’ model fits in the palm of his hand.) You should check it out. It’s pretty sweet.
I think it might be worth mentioning, however, that it’s not necessary to use three-dimensional commutative diagrams to understand the octahedral axiom. There’s a pretty reasonable way of writing it using just 2d diagrams.
If we have the following three exact triangles
then there exists a 4th “vertical” exact triangle such that the following diagram commutes.
Obviously, you can compress this more, since some of the commutative squares are trivial, but I think the content is clearer when the original three triangles are left in their original form. What this axiom says, basically, is that if we identify , , and , then we have .
I think I’ll leave it to the reader to figure out what you have to do to this diagram to make it into an octahedron.
TeXnical Aside: If anyone knows how to make wordpress support xypic I’d appreciate it. At the moment I make commutative diagrams by typing xypic code into the BLANK in
This produces a .gif which I upload.