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Stream: community: general

Topic: 7 or a proper class

John Baez (Oct 12 2020 at 16:45):

A bizarre result by Mark Hovey, reported by Ivo Dell'Ambrogio on the n-Cafe:

Let $k[\mathbb{Z}/2]$ be the group algebra of $\mathbb{Z}/2$ with coefficients in the field $k$. Then:

Theorem. There are precisely 7 closed symmetric monoidal structures on the category of $\mathbb{F}_2[\mathbb{Z}/2]$ -modules. If we replace the field $\mathbb{F}_2$ by a field of characteristic other than 2, there is a proper class of closed symmetric monoidal structures.

Fabrizio Genovese (Oct 12 2020 at 17:17):

Is this somehow related to the "seven trees into one" paper?

Fabrizio Genovese (Oct 12 2020 at 17:18):

https://arxiv.org/pdf/math/9405205v1.pdf

Fabrizio Genovese (Oct 12 2020 at 17:18):

This seems too much of a strange thing to me to appear in two different contexts without any sort of relation

John Baez (Oct 12 2020 at 17:50):

I'd be shocked if they're related. The "seven trees in one" thing is a consequence of $T \cong 1 + T^2$ - a tree is either a root or a root together with a pair of trees - which you can manipulate using ordinary algebra, pretending it's an equation between polynomials, to get $T^7 = T$.

Shea Levy (Oct 12 2020 at 19:37):

John Baez said:

A bizarre result by Mark Hovey, reported by Ivo Dell'Ambrogio on the n-Cafe:

Let $k[\mathbb{Z}/2]$ be the group algebra of $\mathbb{Z}/2$ with coefficients in the field $k$. Then:

Theorem. There are precisely 7 closed symmetric monoidal structures on the category of $\mathbb{F}_2[\mathbb{Z}/2]$ -modules. If we replace the field $\mathbb{F}_2$ by a field of characteristic other than 2, there is a proper class of closed symmetric monoidal structures.

Even primes are strange beasts