Dan Oneata and Peter Addor - thanks for the corrections! I made them all.

Anindya wrote:

> could we write the composite natural transformation as \\(\beta\circ\alpha\\) rather than \\(\beta\alpha\\)?

Yes, and that's what I'll do.

People often change their minds about whether or not they want to use a

\\(\circ\\) for these various forms of composition:

1. composition of morphisms in any category \\(\mathcal{C}\\)

2. composition of functors between categories, and also

3. composition of natural transformations between functors from \\(\mathcal{C}\\) to \\(\mathcal{D}\\)

All of these are actually composition of morphisms in some category, namely

1. any category \\(\mathcal{C}\\)

2. the category \\(\mathbf{Cat}\\)

3. the functor category \\(\mathcal{D}^\mathcal{C}\\)

So, there's an argument to be made that we should use the same notation for all three... and to save space we might as well skip the \\(\circ\\) and just write things like \\(g f\\), \\(G F\\) and \\(\beta \alpha\\).

However, sometimes it's important to use more than one notation for composition. The reason is that there's a second way to compose natural transformations, besides the one I explained in this lecture! It's called 'horizontal composition'. There are also two ways to compose a functor and a natural transformation. Organizing all 5 operations on functors and natural transformations in a clear way turns out to demand two different symbols for composition. When we're done, we say \\(\mathbf{Cat}\\) is a [strict 2-category](https://en.wikipedia.org/wiki/Strict_2-category).

I've been wishy-washy so far, since we're not at this level of sophistication in the course yet, and I'm sort of hoping we'll never get there! This more sophisticated stuff will be lurking in the background when we discuss Kan extensions in a few days, but I'll try to hide it.

But I should at least make a try at being consistent. For some idiotic reason I started out by writing composition using \\(\circ\\). So, for now, I'm going to use \\(\circ\\) for the 3 forms of composition listed above. At some point I may drop the \\(\circ\\), because it gets tiring... but I'll try to remember to warn people when I do this!

Anindya wrote:

> could we write the composite natural transformation as \\(\beta\circ\alpha\\) rather than \\(\beta\alpha\\)?

Yes, and that's what I'll do.

People often change their minds about whether or not they want to use a

\\(\circ\\) for these various forms of composition:

1. composition of morphisms in any category \\(\mathcal{C}\\)

2. composition of functors between categories, and also

3. composition of natural transformations between functors from \\(\mathcal{C}\\) to \\(\mathcal{D}\\)

All of these are actually composition of morphisms in some category, namely

1. any category \\(\mathcal{C}\\)

2. the category \\(\mathbf{Cat}\\)

3. the functor category \\(\mathcal{D}^\mathcal{C}\\)

So, there's an argument to be made that we should use the same notation for all three... and to save space we might as well skip the \\(\circ\\) and just write things like \\(g f\\), \\(G F\\) and \\(\beta \alpha\\).

However, sometimes it's important to use more than one notation for composition. The reason is that there's a second way to compose natural transformations, besides the one I explained in this lecture! It's called 'horizontal composition'. There are also two ways to compose a functor and a natural transformation. Organizing all 5 operations on functors and natural transformations in a clear way turns out to demand two different symbols for composition. When we're done, we say \\(\mathbf{Cat}\\) is a [strict 2-category](https://en.wikipedia.org/wiki/Strict_2-category).

I've been wishy-washy so far, since we're not at this level of sophistication in the course yet, and I'm sort of hoping we'll never get there! This more sophisticated stuff will be lurking in the background when we discuss Kan extensions in a few days, but I'll try to hide it.

But I should at least make a try at being consistent. For some idiotic reason I started out by writing composition using \\(\circ\\). So, for now, I'm going to use \\(\circ\\) for the 3 forms of composition listed above. At some point I may drop the \\(\circ\\), because it gets tiring... but I'll try to remember to warn people when I do this!