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I requested some more details at strict 2-category. It would be nice to have something describing how objects are categories, morphisms are ??? satisfying ???, 2-morphisms are ??? satisfying ???.
I'm sure all the details could be unwrapped from the simple statement "a strict 2-category is a Cat-category", but then I need to learn what an enriched category is first and then I need to see how that works in the case of Cat-enriched category. Soon, I feel overwhelmed. A strict 2-category is probably not THAT hard to understand explicitly.
It looks like Finn Lawler is writing this for you, Eric; if he doesn't finish it, then I will. (But you're right, it's not that hard, so I'm sure he will.) In the meantime, try Wikipedia, whose 2-categories are strict by default.
Yup: just a short bit spelling out some of the definitions. I've dashed this off at home without references, so there may be some mistakes.
By the way, on
... how objects are categories...
they're not, in general. In the 2-category Cat they are, but arbitrary 2-categories, like their 1-cousins, are not required to have any particular structure on their objects.
Ack! Bigons! I do not like bigons! :D
Is there a definition of 2-category that does not rely on bigons?
What have you got against bigons? The 2-cells in a 2-category (strict or weak) are bigons; that's basically the definition of a 2-category. If you want some other shape of cell, then you've got something other than a 2-category.
It's true that some definitions of n-category use cells of other shapes. However, once you have a composition operation that applies to all 1-cells, then a cell of any other shape can be regarded as a bigon by just composing up its source and target. So bigons are the most concise way to describe the structure, and the essential aspect of it, although it is sometimes convenient to also include cells of other shapes in order to describe the composition operations cleanly.
OK, I think that I've filled in all of the details.
And yes, Eric, the usual notion strict 2-category is an inherently globular (bigonal) concept; you can call it a globular strict 2-category if you want to make that precise, but it's the usual default.
There are, however, also simplicial and cubical strict 2-categories, and the weak notion of bicategory (while usually also defined globularly) is indifferent to the shapes used. Urs has considered these matters, mostly for omega-categories, at geometric shapes for higher categories.
I asked a question related to this shape issue at strict 2-category.
I added a corresponding sentence below the query box.
Generally, I'd say the answer to "Should we make xyz more explicit?" is always "Yes!"
Thanks! I asked another question :)
Okay, I have replied again.
I tried to reply in such a way that you can remove the query box if you feel the question has been answered and we are left with proper entry text.
I have now put in details at bicategory to match the details at strict 2-category. I’m not sure that it was worth it, but there it is.
I have now put in details at bicategory to match the details at strict 2-category. I’m not sure that it was worth it, but there it is.
Thanks, Toby. I think it’s worth it. The nLab has or had some curious gaps when it came to the basic definitions of what the central object of interest here is supposed to be. I am glad seeing these eventually being filled.
Curoously enough, I was just yesterday thinking how a category theory resource as nlab has so little explicit standard detail in bicategory entry. Minerva was listening!
I added some more to strict 2-category: some details on the relation to sesquicategory, a bit of history, and some references.
As spurred by the MO discussion here.
Thank you! I might have gotten around to it too, but I’m glad you did.
added to strict 2-category two technical terms and a reference.
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