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Stream: theory: category theory

Topic: flat and sharp notation for adjuncts

Tim Hosgood (Jan 13 2023 at 14:19):

Sometimes people like to use $\flat$ and $\sharp$ for adjuncts, but there seem to be two opposite conventions:

• given $f\colon Lx\to y$ and $g\colon x\to Ry$, the nLab ([[adjunct]]) writes $g=f^\sharp$ and $f=g^\flat$ (so "flats on the left, sharps on the right")
• Category Theory in Context (p. 116) writes $f^\sharp\colon Lx\to y$ and $f^\flat\colon x\to Ry$ (so "sharps on the left, flats on the right")

I know that convention is convention, but is there a reason to prefer one over the other? I guess in cohesion, the flat modality is left adjoint to the sharp modality, so maybe this is why the nLab shows this preference.

Cole Comfort (Jan 13 2023 at 14:36):

In music theory flat means "lower in pitch" and sharp means "higher in pitch". Take the inclusion $\iota: \mathbb{N} \to \mathbb{R}$, then $\lceil - \rceil \dashv i \dashv \lfloor - \rfloor$. So in this example it seems like it would be more appropriate to call the left adjoint to $\iota$ sharp and the right adjoint flat because the floor is lower and the ceiling is higher.

Paolo Perrone (Jan 13 2023 at 14:38):

I suppose you mean N instead of R, right?

Cole Comfort (Jan 13 2023 at 14:38):

Yes, obviously haha

Paolo Perrone (Jan 13 2023 at 14:39):

nope, the other one :p

Cole Comfort (Jan 13 2023 at 14:40):

Suffering from jetlag

Paolo Perrone (Jan 13 2023 at 14:41):

By the way, that's a good example. I never thought that the inclusion of black keys into the whole keyboard are an instance of that! (Approximating sharp and flat to be idempotent, which is not the convention in music.)

Cole Comfort (Jan 13 2023 at 14:56):

Since these were the first examples of adjunctions in my first category theory course, I just assumed the notation came from this.

Mike Shulman (Jan 13 2023 at 15:38):

On the other hand, when we consider posets as categories, the relation $\le$ becomes an arrow $\to$. So in general, it seems that "lower" things are in the domain and "higher" things in the codomain.

Mike Shulman (Jan 13 2023 at 15:39):

The reason for the choice of $\flat$ and $\sharp$ in cohesion, by the way, is that then $\flat BG$ is the classifier for $G$-bundles with a flat connection, in the geometric sense of "no curvature".