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• CommentRowNumber1.
• CommentAuthorEric
• CommentTimeNov 6th 2009

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.

• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeNov 6th 2009

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.

• CommentRowNumber3.
• CommentAuthorFinnLawler
• CommentTimeNov 6th 2009

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.

• CommentRowNumber4.
• CommentAuthorEric
• CommentTimeNov 6th 2009

Ack! Bigons! I do not like bigons! :D

Is there a definition of 2-category that does not rely on bigons?

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeNov 6th 2009

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.

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeNov 6th 2009

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.

• CommentRowNumber7.
• CommentAuthorEric
• CommentTimeNov 16th 2009

I asked a question related to this shape issue at strict 2-category.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 16th 2009

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!"

• CommentRowNumber9.
• CommentAuthorEric
• CommentTimeNov 16th 2009

Thanks! I asked another question :)

strict 2-category

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeNov 16th 2009
• (edited Nov 16th 2009)

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.

• CommentRowNumber11.
• CommentAuthorTobyBartels
• CommentTimeApr 11th 2010

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeApr 11th 2010

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.

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeApr 11th 2010

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!

• CommentRowNumber14.
• CommentAuthorTodd_Trimble
• CommentTimeJul 15th 2016

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.

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeJul 15th 2016

Thank you! I might have gotten around to it too, but I’m glad you did.

• CommentRowNumber16.
• CommentAuthorPeter Heinig
• CommentTimeJul 18th 2017

added to strict 2-category two technical terms and a reference.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 28th 2021

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeAug 29th 2021

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

What’s a good reference, if any, to point a general audience to for strict (2,1)-categories, hence $Grpd$-enriched categories?

All intro/textbook references I have scanned so far speak only of strict 2-categories/$Cat$-enriched categories; while research-level articles that implicitly deal with $Grpd$-enriched categories don’t make the concept explicit.

Not that there is any subtlety in restricting the $Cat$-enrichment to the $Grpd$-enrichment, but the lay person will still appreciate this being made explicit. If nothing else, it saves them from swallowing the usual weasel clauses on size issues.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeAug 30th 2021

ah, this here is not too bad:

• P. H. H. Fantham, E. J. Moore, Groupoid Enriched Categories and Homotopy Theory, Canadian Journal of Mathematics 35 3 (1983) 385-416 (doi:10.4153/CJM-1983-022-8)
1. Hi Urs! I don’t know a nice reference either, but it seems $\mathsf{Grpd}$-enriched categories are also called “track categories” and searching for this name turns a number of potentially nice references.

• CommentRowNumber22.
• CommentAuthorTim_Porter
• CommentTimeSep 1st 2021
• (edited Sep 1st 2021)

The references that Théo gives are good. The terminology ‘track category’ is current in the work of Hans Baues and his collaborators, including some of Hans’ books.

BTW there is also:

2-Groupoid Enrichments in Homotopy Theory and Algebra, K.H.Kamps, and T.Porter K-Theory v. 25, n. 4 (April, 2002): 373-409., ;-)

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeSep 1st 2021

Thanks. I remember seeing the “track”-terminology from when I was looking at those references doing Toda-brackets as pasting diagrams in homotopy 2-categories (here).

So I have added a pointer at strict (2,1)-category to a book by Baues (here) for this alternative terminology. Unfortunately, besides introducing alternative terminology, that book does not pause a moment to recall what a Grpd-enriched category actually is. Same for followups that I have seen so far.

• CommentRowNumber24.
• CommentAuthorTim_Porter
• CommentTimeSep 1st 2021

Hans Baues’ books often go in quite deeply quite quickly!

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeSep 1st 2021

Most references are like this. That’s why I am asking (#19) for those that are not.