2-categories 1 (Catsters 36)

The definition of (strict) 2-category.

Category:

Howto & Style

Tags:

• categories
• 2-categories

[ggnt]

[ctd:2]Granted, you do give a motivation here before launching into the defn, but I guess I am looking for something like "deriving the definition", i.e., given an idea of the thing we are after, "discovering" the definition, kind of like the first guy who did it would.

I recall wanting to make a similar comment on your lecture on natural tfms. I realize much of what I say here is subjective, so I won't be surprised if you already thought about all this and just disagree with what I say.

[TheCatsters]

Thanks for your comments. It's true that different people learn and understand things in different ways. We make the videos we feel like making, in the hope that they will help someone. We probably explain things in the way that we understand them, as that will come out best. Perhaps someone else will make some videos explaining things a different way!

[ggnt]

Thank you, this was helpful. A quick question: Instead of defining 2-cells as arrows between morphisms f,g that are both between the same pair of objects x-->y, why don't we allow f and g to be between different objects, as in alpha:f-->g, with f:x-->y and g:z-->w? It seems like it would still be possible to define horizontal and vertical compositions of 2-cells if we have this freedom, but maybe I'm being naive?

[TheCatsters]

One answer would be that it's not so useful for the examples we have in mind. Another is that this is the way things naturally arise if you look at, for example, categories enriched in categories.

[aurinkona]

oh god... "zed"... oh noooooo BRITS! :]

[ggnt]

[ctd]I find it much easier to understand a definition when someone first describes how one arrives at it by a nice motivation. It seems to me that I can even come up with the definitions myself when things are introduced this way. If this order is reversed, i.e., if the rigorous definition is given first, and elaborated later, I get confused in the beginning, have a hard time concentrating on the following elaborations. Many rigorous definitions are totally mystifying the first time you see them

[ggnt]

A quick (perhaps subjective) comment on pedagogical style: I've watched a couple of your lectures; while I find them very "clean" and helpful, there is one characteristic that sometimes makes it hard for me to go into the guts of the subject easily. This is hard to describe, but it is perhaps similar to defining the derivative as a (rigorous) limit first, and then explaining what this means intuitively/geometrically. [Continued in the next comment]