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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.
added pointer to:
added pointer to:
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.
ah, this here is not too bad:
[deleted]
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., ;-)
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.
Hans Baues’ books often go in quite deeply quite quickly!
Most references are like this. That’s why I am asking (#19) for those that are not.
In revision 25, Todd Trimble added the phrase “Ehresmann, who in fact defined double categories, and 2-categories as a special case”. Can anyone give an explicit reference for the fact that Ehresmann defined 2-categories? 2-categories are a special case of double categories, but this does not imply that Ehresmann identified them explicitly, and I cannot find a reference in Ehresmann’s paper.
added pointer to:
Re #27: I don’t know where I got that specific statement from, but Wikipedia does report that Ehresmann defined 2-categories in 1965 (apparently not in 1963 when he introduced double categories), as an example of an enriched category. I’ll add the reference in a moment.
For the attribution to Ehresmann we ought to add a concrete pointer to the page number and/or the definition number.
(Once I tried to find the definition of “internal category” in the similarly titled “Catégories structurées”, which many people cite as the origin of the concept, and I got away with the impression that it’s not really there.)
Now, I haven’t found the book “Catégories et structures” online yet, but there is this:
If the definition of strict 2-category is in there, then let’s say on which page exactly. If not, and if we can’t give any page number, then we should maybe add a cautionary remark that the attribution is not verifiable.
I spent some time looking for the definition of a 2-category in Ehresmann’s work previously, and could not find it. However, I did not (and still do not) have access to the book “Catégories structurées”, so it may be defined there. I agree that it would be good to confirm this.
I have the impression, just reading the introduction of “Catégories structurées”, that what Ehresmann calls “catégories $\mathcal{H}$-structurées” are what we would call in English an “internal categories in $\mathcal{H}$”. Is that a wrong impression?
(I admit that I have trouble reading Ehresmann quickly, probably mostly because my French is weak, but also because of the style in which he writes.)
I came to suspect that nobody had the patience to penetrate what Ehresmann writes, so that some concepts got attributed to him just due to exhaustion of the readers.
But I’d be happy to be proven wrong about this.
If there is a recognizable definition in “Catégories structurées”, I’d like to add the explicit pointer (best a scan of the respective paragraph) to the list of historical references that I once tried to collect at internalization and at internal category. On the other hand, for internal categories (and internalization in general) the transparent definition by Grothendieck – which already is our modern definition – anyways predates “Catégories structurées” by a couple of years – these concepts are Grothendieck’s not Ehresmann’s.
On the other hand, if we do not find the definitions in Ehresmann’s work then we should not perpetuate the suggestion that he stated them.
So for the time being I have moved “Catégories et structures” out of the list of “original articles” and instead added this remark:
(Wikipedia asserts that the definition of strict 2-categories is also due to Charles Ehresmann’s Catégories et structures (Dunod, Paris, 1965), but so far we have been unable to identify it there.)
Of course, the same comment applies to the claim that the notion is due to Godement’s “Topologie algébrique et theorie des faisceaux”. I haven’t seen that article either. There are reviews of it by Whitehead (doi:10.2307/3608544), by Simms (doi:10.1017/S0013091500021799) and by Buchsbaum (here) and neither mention an occurrence of 2-categories.
[edit: I see that no 2-categories are meant to be in Godement’s article, just a formulation of “five rules of functorial calculus” in the appendix. Since Godement won’t have written the English words “five rules of functorial calculus”, what is the exact French phrase he did use?]
I think the only assertion is that Godement had formulated his “five rules of functorial calculus”, and the notion of 2-category abstracts his rules in axiomatic form.
I still have tabs open for finding the relevant material in Ehresmann, but I haven’t looked much into it since May 20. I’d bet money you won’t find him using the term “2-category”, so I think what’s going on with other authors like Street, Wikipedia, many others, is that the essential idea is to be found in his work using different words, to the effect of “$Cat$-structured categories” and the like, and that some license allowing for that fact is due.
Yes, but we should not propel rumours.
It also worries me that everyone claims Godement formulated “five rules of functorial calculus” without ever mentioning the actual French phrase that Godement must have used instead. Best would be to get hold of the article and make a scan.
I understand. So one source that alleges Godement’s five rules is the opening paragraph of Eilenberg-Kelly, A Generalization of the Functorial Calculus, where they direct the reader to the appendice. I place a fair amount of confidence in the scholarship of Eilenberg and Kelly, but yes, it would be nice to lay eyes on it.
I think maybe the Godement work is a book, not an article? Anyway, it (Theorie des Faisceaux) might not be the same as Topologie algébrique et theorie des faisceaux. The former is dated 1958, whereas that webpage may be for a 1997 edition of something, and who knows what may be different between the two.
Ah, found it here, page 281 (“cinq règles de calcul fonctoriel”). I’d make a scan, but I don’t know how to do that easily.
Taking down some notes here. I refer to his Collected Works, in particular the first article given in Volume III.1 here, cited as C.R.A.S. tome 256 (Séance du 4 février 1963), p. 1198-1201. Here Ehresmann gives a clear definition of double category (CATÉGORIES DOUBLES), with the same meaning we give for double category. Here is my free translation:
By a double category we mean a class $\mathcal{C}$ equipped with two laws of composition, denoted $\cdot$ and $\perp$, satisfying the following conditions:
$(\mathcal{C}, \cdot)$ is a category, denoted $\mathcal{C}^{\cdot}$; the right and left identities of $f \in \mathcal{C}^{\cdot}$ will be denoted $\alpha(f)$ and $\beta(f)$ respectively, the class of identities, $\mathcal{C}_0^\cdot$;
$(\mathcal{C}, \perp)$ is a category, denoted $\mathcal{C}^\perp$; the identities of $f \in \mathcal{C}^\perp$ will be denoted $\alpha^\perp(f)$ and $\beta^\perp(f)$, the class of identities, $\mathcal{C}_0^\perp$;
The maps $\alpha$ and $\beta$ (resp. $\alpha^\perp$ and $\beta^\perp$) are functors from $\mathcal{C}^\perp$ to $\mathcal{C}^\perp$ (resp. from $\mathcal{C}^\cdot$ to $\mathcal{C}^\cdot$);
Interchange axiom: if the composites $k \cdot h$, $g \cdot f$, $k \perp g$ and $h \perp f$ are defined, then one has
$(k \cdot h) \perp (g \cdot f) = (k \perp g) \cdot (h \perp f).$It’s clear that he’s using a “morphisms-only” (i.e., one-sorted) definition of category, so that his $\mathcal{C}$ should be thought of as the class of 2-cells, but with that in mind, it should be understandable. I’m still trying to get hold of his 1965 book.
People reading this comment might be amused by Linton’s scathing review of this book in the American Mathematical Monthly, doi:10.2307/2314770. I do see a caution, “2-categories appear under the name double category”. Hmm… now I really want to see it!
Finally, I have added hyperlinked to a bunch of terms, such as 1-morphism, 2-morphism, horizontal composition, vertical composition, whiskering and interchange law. Also cross-linked with enriched groupoid.
For what it’s worth, when I attempted to find the introduction of 2-categories before, the earliest reference I found was in Bénabou’s 1965 Catégories relatives (see Example 5), where he cites his “forthcoming” thesis Algèbre élémentaire dans les catégories, and tells the reader to see also Ehresmann’s Catégories structurées (though he doesn’t explicitly say 2-categories appear there, so he may simply be referencing the work for the similar definition of double categories). Bénabou’s thesis was eventually completed as Structures algebriques dans les categories, where the term “2-catégorie” does appear.
Perhaps it would be worth asking Andrée Ehresmann directly, as she would surely know whether Charles Ehresmann ever defined 2-categories explicitly? It would be nice to find a copy of Catégories et structures also, though.
It does seem plausible to me that it was actually Bénabou who introduced 2-categories.
also added pointer to
added pointer to
(thanks to varkor here)
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