2-morphism
A 2-morphism in an n-category is a k-morphism for : it is a higher morphism between ordinary 1-morphisms.
So in the hierarchy of -categories, the first step where 2-morphisms appear is in a 2-category. This includes cases such as bicategory, 2-groupoid or double category.
There are different geometric shapes for higher structures: globes, simplices, cubes, etc. Accordingly, 2-morphisms may appear in different guises:
A globular -morphism looks like this:
A simplicial -morphism looks like this:
A cubical -morphism looks like this:
Of course, using identity morphisms and composition, we can turn one into the other; which is more fundamental depends on which shapes you prefer.
Eric: Are there any consistency requirements for a 2-morphism? For example, in the bigon above, if , , and , are there requirements on regarding and ? For example, should come with component 1-morphisms and such that
or maybe
? Could there be a 2-morphism without the corresponding 1-morphism components?
Urs Schreiber: in any given 2-category you have to specify which 2-morphisms exactly there are supposed to be, what exactly you allow between and . When you ask about components, it seems you are thinking of 2-morphisms specifically in the 2-category Cat. Here, yes, the allowed 2-morphisms are those that are natural transformations between their source and target 1-morphisms, which are functors.
Eric: I think the exchange law might be what I had in mind.
In the 2-category Cat, 2-morphisms are natural transformations between functors.
In a path 2-groupoid 2-morphisms are certain surfaces or images of surfaces in a space, going between paths in that space.
Last revised on May 25, 2016 at 14:11:41. See the history of this page for a list of all contributions to it.