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Stream: deprecated: mathematics

Topic: are monads monads?

Tim Hosgood (Mar 16 2022 at 13:10):

If you ask a (certain type of) algebraic geometer what a monad is, they will give you the following definition:

A monad $M$ on a projective variety $X$ is a short sequence of coherent sheaves $0\to\cal{F}\to\cal{G}\to\cal{H}\to0$ on $X$ that is exact at $\cal{F}$ and $\cal{H}$ (i.e. such that $\cal{F}\to\cal{G}$ is injective, $\cal{G}\to\cal{H}$ is surjective, and the composite is zero).

Is there any way in which this is an example of a monad in the sense of category theory? Or is this just a really terrible unfortunate clash of terminology?

Joe Moeller (Mar 16 2022 at 13:17):

Is the composite $\mathcal F \to \mathcal G \to \mathcal H$ zero? Wouldn't that be exactness at $\mathcal G$? Maybe I'm just confused.

Tim Hosgood (Mar 16 2022 at 13:20):

exactness at $\cal{G}$ would say that the kernel of $\cal{G}\to\cal{H}$ is exactly the image of $\cal{F}\to\cal{G}$ (i.e. that the cohomology is zero), but here we're just asking for the kernel to be contained inside (and not necessarily equal to) the image

Reid Barton (Mar 16 2022 at 13:22):

What area of algebraic geometry is this term/notion used in?

Tim Hosgood (Mar 16 2022 at 13:23):

a lot of projective geometry, e.g. https://arxiv.org/abs/1801.00151

Tobias Fritz (Mar 16 2022 at 14:06):

A quick search suggests that the term goes back to Horrocks: Vector Bundles on the Punctured Spectrum of a Local Ring, where it is introduced at the bottom of p.698. It seems that no explanation for the term is given, but maybe it's motivated by the usage of "monad" as meaning infinitesimal neighbourhood?

Tim Hosgood (Mar 16 2022 at 14:20):

ah, that's an unfortunate (and disappointing) coincidence then

Tim Hosgood (Mar 16 2022 at 14:20):

i was hoping to have my mind blown learning how this notion of monad was secretly a monad...

Tobias Fritz (Mar 16 2022 at 14:30):

Who knows, maybe there nevertheless is an amazing coincidence about monads being monads? As with what happened with 'spectrum": the fact that optical spectra are spectra of operators is a mind-blowing coincidence, especially given that both terms were introduced before this connection was known! (Although the mathematical term may have been inspired by the physics parlance.)

John Baez (Mar 16 2022 at 18:09):

I wonder who came up with this usage of "monad" and when. That might be a clue. Meanwhile I'll try to think about how to get an actual monad from this things.

Todd Trimble (Mar 17 2022 at 13:48):

See monad, but to answer the question, I think it might have been Leibniz.

Jacques Carette (Mar 17 2022 at 19:31):

Leibniz's notion of Monad is much more closely related to Yoneda that the modern notion of Monad.