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Stream: learning: questions

Topic: Notation for (un)currying

Ralph Sarkis (Apr 09 2021 at 10:12):

I am looking for a nice notation for currying and uncurrying that is not too verbose (i.e.: not $\mathrm{curry}(f)$ and $\mathrm{uncurry}(f)$) but still relatively specific (i.e.: not $f'$). By (un)currying, I mean the general version: an isomorphism $\mathrm{Hom}_{\mathbf{C}}(B, A^X)\cong \mathrm{Hom}_{\mathbf{C}}(B\times X, A)$.

Right now, I am using $\lambda f$ and $\lambda^{-1}f$ for currying and uncurrying respectively (it is specific because I don't use lambda calculus notation in my document), it is OK, I am looking for better. Before, I was using $f^{\mathsf{c}}$ and $f_{\mathsf{c}}$, but I don't like the fact that it hints at currying being covariant and uncurrying being contravariant.

Dylan Braithwaite (Apr 09 2021 at 11:08):

I've seen $f^\wedge$ and $f^\vee$ used for transposing along a general adjunction. The nlab page for "adjunct" suggests $f^\sharp$ and $f^\flat$ too. Maybe these are still overly general?

Ralph Sarkis (Apr 09 2021 at 11:13):

Dylan Braithwaite said:

Maybe these are still overly general?

Yes. I am using $f^{\mathrm{t}}$ for both transpositions in an arbitrary adjunctions, but they are introduced after my currying examples.

Jules Hedges (Apr 09 2021 at 11:42):

Personally, I often write it without any notation, I just abuse notation and write $f (x, y) = f x y$, and $f (x, -) = f x$ for the partial application. This works fine for routine stuff, but there's probably cases where it's a bad idea

Ralph Sarkis (Apr 09 2021 at 13:13):

Yeah, my first option was to have a notation to distinguish the two when I am explaining currying and then omit it later on if it is too cumbersome. I am looking for a light and clear notation so that I don't have to omit it.

Mike Shulman (Apr 09 2021 at 14:56):

I don't know how much lighter and clearer you can get than $\lambda f$ and $\lambda^{-1}f$.

John Baez (Apr 09 2021 at 15:22):

I sometimes use $\tilde f$ for currying $f$, and an under-tilde for uncurrying.

Tom Hirschowitz (Apr 10 2021 at 09:12):

You could perhaps try to work out a horizontally symmetrised lambda?

Ralph Sarkis (Apr 10 2021 at 09:33):

Good idea! One reason I don't love plain $\lambda$ is that I am explaining the isomorphism of categories $[C,[D,E]] \cong [C\times D,E]$, so I am applying $\lambda$ to natural transformations which are also greek letters ($\phi$ and $\eta$). I think I'll use $\mathsf{\Lambda}$ or $\mathtt{\Lambda}$.