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New page at sesquicategory.
What’s the reason for this name? (See my addition to the Remarks.)
I presume the reason is that it’s “part of the way” from a 1-category to a 2-category; you add the 2-cells and some of the structure, but you leave out one of the axioms.
The second and third definitions given there are a bit tricky if you want to define a weak sesquicategory, yes? Since such need not have an underlying 1-category.
I would presume that, except that to mind it goes the wrong way. $2$-categories are more general than $1$-categories, and sesquicategories are more general still.
However, I can see that from a strict perspective, it could go as you say. A strict $2$-category has an underlying $1$-category, etc.
I’ve added a fourth definition, which I unaccountably omitted last night, in which a sesquicategory is a category enriched in Cat with the other tensor product (whose corresponding internal hom gives categories of ’unnatural’ transformations). Apparently this and the usual one are the only monoidal closed structures on Cat. I haven’t checked yet, but maybe this is the right way to define weak sesquicategories.
I’ve also added a link to a cat-list discussion that explains the origin of the name.
Toby, while it’s true that 2-categories are a generalization of 1-categories, I don’t usually think of them that way. Instead I think of them as 1-categories with additional extra structure (or stuff, or whatever), which is also true. From that perspective, sesquicategories do sit in between 1-categories and 2-categories.
I think of them as 1-categories with additional extra structure (or stuff, or whatever), which is also true.
This is only true for strict $2$-categories. In contrast, thinking of $1$-categories as $2$-categories with an extra property works both strictly and weakly; and furthermore it is a very important perspective when thinking about higher categories. (All of Lurie’s and Urs’s work in generalising from $1$-categories to $(\infty,1)$-categories implicitly takes this perspective.)
None of this contradicts what you’ve said, but I’m honestly surprised that you usually think of (even strict) $2$-categories in this way!
By the way, I am not suggesting changing the name; I just wanted to see where it came from. It’s a name that I very much doubt will generalise, but it uses grammar which would be hard to generalise anyway. If we later come up with something that really deserves the name $1\frac{1}{2}$-category, then we can still ignore the Latin and just use that.
I’ve edited the Remark just a bit.
This is only true for strict 2-categories.
It depends on what forgetful functor 2Cat -> Cat you are thinking of. It’s true that only strict 2-categories have an underlying 1-category obtained by simply discarding the 2-morphisms, but weak 2-categories still have a “homotopy” 1-category obtained by identifying isomorphic 1-morphisms and then discarding the 2-morphisms. The first functor is right adjoint to the inclusion of 1-categories into strict 2-categories; the second is left adjoint if you restrict it to (2,1)-categories.
What I wrote in #6 is an exaggeration; I think both perspectives are important. But not infrequently I find myself doing things with 2-categories that are analogous to things that one does with 1-categories, but do not reduce to their 1-categorical correspondents when applied to locally discrete 2-categories, and I would argue that the same is true of Lurie and Urs. For instance, a 1-topos, regarded as a locally discrete 2-category, is not a 2-topos.
I don’t really like the name “sesquicategories” either, but for a different reason: regardless of whether they are more or less general than 2-categories, their difference from 2-categories is not really in an “up-or-down” direction!
But not infrequently I find myself doing things with 2-categories that are analogous to things that one does with 1-categories, but do not reduce to their 1-categorical correspondents when applied to locally discrete 2-categories, and I would argue that the same is true of Lurie and Urs.
Sure. But in those cases, one should change the name of the concept, because it is not the same concept but merely an analogous one. Whereas, when they do so reduce, then we keep the old name, because now it is the same concept applied in a more general context.
I don’t really see how the issue of renaming or not is relevant to the question of how 2-categories are related to 1-categories?
I think sesquicategory is somewhat standard term in Australian category theory school. It has enough of general feeling that it is remembrable. I mean it reminds to roughly the right thing. So why should we play Bourbaki and correct the Australians (as if most of the terminology in mathematics were entirely logically ordered) ?
I don’t think anyone was proposing to rename it, just having fun complaining about the existing name. (-:
12 :) That was instructive, I mean all the comments…
@ Zoran #12:
Agreed, let’s not rename it.
@ Mike #10:
It’s when things are not renamed that we see that people think of things as being (not merely analogous but) the same, in the sense of being special cases (possibly one of the other, possibly both of some more general situation). We abandon the historical terms ‘$2$-functor’ and ‘bifunctor’ for ‘functor’, since we recognise a functor between $1$-categories as a special case of a functor between $2$-categories, so the latter is the same thing as the former, deserving of the same name. In contrast, things which are merely analogous get new names.
I agree that both perspectives are useful. But I have a pretty clear idea in my mind which one is primary, and I’m surprised that you have a different idea, especially given that you were the one who clarified for me how the names ought to work.
We abandon the historical terms ‘2-functor’ and ‘bifunctor’ for ‘functor’, since we recognise a functor between 1-categories as a special case of a functor between 2-categories
That’s not what I would say. I would say that we abandon those terms because a functor between 2-categories, as we mean it, is the most appropriate notion of morphism between 2-categories, analogously to functors between 1-categories. It happens that when we regard 1-categories as particular 2-categories, functors of 2-categories reduce to functors of 1-categories, and that’s certainly a good thing that avoids confusion, but it’s not a prerequisite to usage of the unqualified “functor.”
There are other situations in which we use unqualified words in a generalized context that don’t reduce to the previous notions when specialized. For instance, a span between two groupoids does not reduce to a span between two sets when the groupoids are discrete. A profunctor between 2-categories does not reduce to a profunctor between 1-categories when the 2-categories are discrete. And even with ordinary functors, for an arbitrary enriching category V, a functor between V-categories need not reduce to an ordinary functor when the V-categories are “discrete.”
especially given that you were the one who clarified for me how the names ought to work.
What are you referring to? I want to know if I’m being inconsistent… (-:
Your first paragraph is again quite surprising to me! Maybe I’m just very mixed up about what you used to say; in the extreme case, maybe I’m mixing you up with somebody else (but I’m sure that it was you, really).
For instance, a span between two groupoids does not reduce to a span between two sets when the groupoids are discrete.
True, but this is because ‘span’ is an extremely general term that makes sense in any $\infty$-category, here applied to $Grpd$ and $Set$. You’re right that it’s an exception to the rule that I stated, however. (So is the profunctor example, although I never heard you apply this —and never meant to extend it myself— to general enriched categories.)
What are you referring to?
It’s spread out over naming discussions that are organised by the thing being named, not the principle at play, but I’ll see if I can find something definitive.
OK, you state the principle at n-prefix (michaelshulman). I’ll quote the entire second paragraph:
If X has a meaning for 1-categories, then if X is used without a prefix for 2-categories it should include the existing notion for 1-categories as a special case (when 1-categories are considered as homwise-discrete 2-categories). If we consider a notion related to X but which is not a “conservative categorification” in this sense, we will call it 2-X; cf. subcategory (nlab). For instance, we say regular 2-category since a 1-category is regular as a 2-category iff it is regular as a 1-category, but 2-exact 2-category since an exact 1-category is almost never 2-exact as a 2-category.
For the cited discussion at subcategory, it’s probably better to read a forum comment by you. While functions have images, functors have both $1$-images and $2$-images; as you explain, we know which is which by using the convention that the $1$-image of a functor between discrete categories is the same as the image of the corresponding function between sets. You then apply this to the term ‘subcategory’. (So you don’t limit this principle only to your exposition of $2$-topos theory.)
Also explained at that comment (which I had forgotten) is that it’s important to write $\mathbf{B}G$ when interpreting $G$ as a category, in part because the numbering doesn’t correspond in this case. (And indeed, a group is not a category with extra property but a category with extra structure, since it’s really a pointed category with extra property, and $\mathbf{B}G$ is simply its underlying category, an observation also due to you. So in both cases, we get the rule that terms are preserved when interpreting an A as a B with extra property but not when interpreting an A as a B with extra structure.)
It is reading all that, as well as other applications of the same principle, that made me unprepared for #6.
Hmm, interesting. Possibly I’ve gotten looser with terminology over time, although I still do agree with that principle in general. But I also don’t feel a contradiction between wanting terminology for 2-categories to reduce to existing terminology for locally discrete ones, but sometimes regarding 2-categories as 1-categories with extra stuff. In fact, of course, it often happens that “an X with extra stuff” is a generalization of an X, if we can take the extra stuff to be “trivial” in some canonical way. Though I can see that there is some interesting interplay.
Possibly I’ve gotten looser with terminology over time
Possibly. But I realised that the principle was fresh in my mind when you wrote the surprising comment #6 because you had (quite correctly) chastised me with it barely a week earlier here.
I also don’t feel a contradiction between wanting terminology for 2-categories to reduce to existing terminology for locally discrete ones, but sometimes regarding 2-categories as 1-categories with extra stuff.
I agree, but “sometimes regarding” is weaker than “usually think”.
but “sometimes regarding” is weaker than “usually think”.
Yes, and I agreed in #8 that #6 was overstated.
That’s right, you did; everything’s OK.
I’ve rearranged the Definition section at sesquicategory, to make it clear that there are essentially two definitions and not four.
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