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Stream: learning: questions

Topic: Notation for 2-cell compositions?

Shea Levy (Oct 31 2020 at 12:48):

Is there a notational way to distinguish horizontal and vertical composition? I feel like I've seen $\circ$ for both.

Fabrizio Genovese (Oct 31 2020 at 12:52):

Some people use $\circ_n$, where $n$ denotes the dimensionality of composition

Fabrizio Genovese (Oct 31 2020 at 12:53):

For a k-cell, you have $\circ_1, \dots, \circ_k$, where $\circ_1$ denotes composition along the $1$-boundary, and so on.

Amar Hadzihasanovic (Oct 31 2020 at 13:28):

Should be $\circ_0, \ldots, \circ_{k-1}$, but yes.

Amar Hadzihasanovic (Oct 31 2020 at 13:30):

If it's just 2-categories you could also use $\otimes$ for horizontal, as in monoidal categories. I think that's what Cockett and Seely use, possibly others too.

fosco (Oct 31 2020 at 16:06):

Shea Levy said:

Is there a notational way to distinguish horizontal and vertical composition? I feel like I've seen $\circ$ for both.

\boxvert, and \boxminus, of course! (I never remember which one is already defined, the one that isn't is just a \mathop of the \rotatebox{90} of the other.

Nathanael Arkor (Nov 01 2020 at 11:27):

Personally, I find it most intuitive to use $\circ$ for horizontal composition, because it's in the same direction as 1-categorical composition, and then one can use some other symbol, e.g. $\bullet$, for vertical composition.

Reid Barton (Nov 01 2020 at 11:47):

This convention would mean that if $\alpha : F \to G$ and $\beta : G \to H$ are natural transformations, the notation for their composition would depend on whether you're currently viewing them as 2-morphisms of Cat or 1-morphisms of a category of functors... which seems like a pretty thin distinction to draw.

Reid Barton (Nov 01 2020 at 11:47):

In practice I think most people would just denote both kinds of composition by juxtaposition and let context disambiguate.

John Baez (Nov 01 2020 at 15:48):

I've always seen people use separate notations, and this seems pretty important, since one often uses both kinds of composition together, e.g. in the interchange law

$(\alpha \circ \beta) \otimes (\alpha' \circ \beta') = (\alpha \otimes \alpha') \circ (\beta \otimes \beta')$

which if one denotes both forms of composition by juxtaposition becomes

$(\alpha \beta) (\alpha' \beta') = (\alpha \alpha') (\beta \beta')$

Reid Barton (Nov 01 2020 at 20:18):

I guess I've also heard of $*$ being used for horizontal composition. It doesn't come up too often, anyways.

Reid Barton (Nov 01 2020 at 20:18):

At the point where the notation starts to become nonobvious, it would be better to have a diagram.

John Baez (Nov 01 2020 at 20:47):

Some days I use horizontal and vertical composition in 2-categories all day long. I've used various notations at various times, since I don't think there's a standard one. The main thing is to explain your notation.

Reid Barton (Nov 01 2020 at 21:24):

I'm thinking of situations in ordinary category theory, like the triangular equations for an adjunction.

Reid Barton (Nov 01 2020 at 21:26):

I'd expect to see something like $1 = \varepsilon F \circ F \eta$, with an explicit $\circ$ for clarity, but otherwise using juxtaposition where possible. I'd be surprised to see $\circ_i$ or any notation which needed explanation.

John Baez (Nov 01 2020 at 21:44):

Right, $\circ_i$ tends to show up when you hit 3-categories or higher.

John Baez (Nov 01 2020 at 21:45):

I use a lot of mixtures of horizontal and vertical composition of natural transformations in category theory, but as you suggest, it's best to describe these using 2-categorical pasting diagrams: they're a lot clearer.