Michael Shulman n-prefix

2-categorical conventions

In line with the rest of the nLab, 2-category means weak 2-category, or precisely bicategory. All notions such as limits, adjunctions, equivalences, and so on are understood in a suitably weak sense (without need for a prefix such as “bi-”). Note that without choice, even Cat is not a strict 2-category, since we must use anafunctors instead of functors; see regular 2-categories and choice.

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. 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.

$n$-categories for $n\le 2$

In many cases it turns out to be just as easy to develop the theory not just for 2-categories, but for $(s,r)$-categories for some $0\le s\le 2$ and $r\gt 0$. We will say that $n\le 2$ and $n$ is directed to mean that $n=(s,r)$ for some such $s,r$. This includes the following cases:

2 = (2,2)

of course, a 2-category is just a 2-category

(2,1)

A (2,1)-category is a 2-category all of whose 2-cells are invertible, such as Gpd.

(1,2)

A (1,2)-category is a 2-category in which any two parallel 2-cells are equal, such as Pos.

1 = (1,1)

A 1-category is a 2-category in which any two parallel 2-cells are equal and invertible; that is, a 2-category which is both a (2,1)-category and a (1,2)-category. This means its hom-categories are equivalent to discrete sets, so it can be identified with a 1-category in the usual sense.

(0,1)

A (0,1)-category is a poset, or more precisely a preordered set.

The following values of $n=(s,r)$ are considered $\le 2$, but are not directed.

(2,0)

A (2,0)-category is a 2-groupoid: a 2-category in which all 2-cells are isomorphisms and all 1-cells are equivalences.

(1,0)

A (1,0)-category is a groupoid.

0=(0,0)

A 0-category is a set (or a category, or 2-category, that is equivalent to one).

-1 = (-1,0)

A (-1,0)-category is poset in which any two objects are isomorphic; thus it is a truth value (the truth value of “there exists an object”). Since there is no useful notion of (-1,-1)-category, (-1,0)-categories are often called (-1)-categories and we will sometimes use this convention.

-2 = (-2,-1)

A (-2,-1)-category is an inhabited (-1,0)-category; that is, it is the truth value “true.” As with (-1)-categories, (-2,-1)-categories are often called simply (-2)-categories.

These values of $n$ are also important, but for them one doesn’t expect a notion of $n$-topos, nor of regularity, exactness, extensivity, or an internal logic. For instance, a nontrivial groupoid has no limits, so it certainly cannot be regular.

Actually, I’ve been mildly surprised not to see you use these values of $n$. Sure, the concepts tend to be trivial then, but so they also tend to be when $n \lt 0$.

And actually, I don't think it's appropriate to refer to $n \lt 0$ as ‘directed’. Directedness is provided only by $r \gt 0$.

—Toby

You’re right about directedness; I’ve fixed the page. What sort of “use” of nondirected $n$ were you expecting? They do come up, of course; for instance it’s important to talk about $n$-truncated objects when $n$ is not necessarily directed. The only point is that no nontrivial $n$-categories can be regular, exact, coherent, etc. unless $n$ is directed.

I guess it might be useful to distinguish $n$-truncated 2-toposes even when $n$ is not directed. For instance, 0-truncated (Grothendieck) 2-toposes are just powers of $Cat$, and (1,0)-truncated 1-toposes are toposes of actions by a groupoid. But you can’t say any more that an $n$-truncated 2-topos is the category of 2-sheaves on an $n$-topos, since there is no longer any notion of $n$-topos.

I incorporated your other comments into the text. -Mike

By ‘use’ I meant at a minimum ‘mention’. That is, remark that no groupoid is regular, etc. Mind you, I've been only mildly surprised not to see this (and such surprise as I had came in part because you sometimes did mention $n \leq 0$); I like to mention degenerate cases, but I know that others don't. Possibly we can regard the paragraph above as covering it now. —Toby

We adopt the natural convention that $(s,r)\pm 1 = (s\pm 1,r\pm 1)$. Thus, for instance, saying that $n$-categories form an $(n+1)$-category includes the statements that:

• sets form a 1-category
• categories form a 2-category
• groupoids form a (2,1)-category
• posets form a (1,2)-category
• truth values form a poset

and so on. For this convention it is important to remember that (-1)-category is a shorthand for (-1,0)-category, so that $(-1)+1 = (0,1)$, not (0,0). And of course, $(-2)+1=(-1)$. Note that $n$ is directed if and only if it is of the form $m+1$ for some $m\ge (-1,0)$.

Note that $(s,r)$-categories are the same as $(s,r')$-categories whenever either $r,r'\ge s+1$ or $r,r'\le 0$. (Although for monoidal categories, $r=-1$ and $r=0$ can be distinguished.) Thus, to avoid duplication, usually one restricts to $0\le r\le s+1$. However, these requirements conflict when $s=-2$ (which is something of a special case in other ways as well). We choose -2 to mean (-2,-1) so that we obtain (-1,0) upon adding 1 to it, although (-2,0) would work just as well as long as we remember that (0,2) and (0,1) are the same. Thus, our general restriction on $r$ and $s$ is $min(0,s+1) \le r \le s+1$.

Discussion

From a previous discussion at truncated object, which doesn't really belong there:

Yes, definitely. A so called ‘$-1$-category’ (a truth value) is really a $-1$-groupoid or a $0$-poset, that is a $(-1,0)$-category, not a $(-1,-1)$-category (which doesn't really make sense) at all. As you know. —Toby

Yeah. But (-1,0)-category is cumbersome to say when there is no different notion of (-1)-category that it needs to be distinguished from. (-: And it seems to break down at (-2) anyway; for the adding-one convention to continue to work we should write (-2,-1)-category instead of (-2)-category, but that doesn’t make sense in the general scheme of (n,r)-category. I suppose we could decide to write $(n,\infty)$ instead of $(n,n+1)$ in all cases… -Mike

Partly, this would work better if we said ‘$n$-poset’ all the time for various values of $n$. In general, an $(r,s)$-category is an $(r+1,s)$-poset. The bad part is that now these pages focus on $n \leq (3,2)$, which is not so slick to say. But the general theory works better; we require $0 \leq s \leq r$ (with the same exception as before, now when $r = -1$) and no glitch crossing from $-1$ to $-2$. —Toby

I suppose so, but I don’t think you’ll have much luck trying to get people to say “(3,2)-poset” instead of “2-category.” (-: -Mike

Nothing wrong with ‘$2$-category’, ‘$2$-groupoid’, etc as abbreviations, but we shouldn't lose sight of the fundamental concept. (Similarly, wasn't someone trying to come up with a general name for $(n,1)$-category?) —Toby

What are you claiming is the “fundamental concept?” The only candidates for a fundamental concept that I see are either full-blown $\omega$-categories or $(s,r)$-categories for arbitrary $s,r$. The only remaining question is terminology, and I think saying that “2-category” is an abbreviation for “(2,2)-category” is much more likely to catch on than saying that it is an abbreviation for “(3,2)-poset.” (In fact, it has already caught on.) In particular, there is a virtue to continuing to use the word “category,” which communicates the behavior of these things much better than “poset.” Already I think “(1,2)-category” is a more descriptive name than “2-poset.” -Mike

I agree with Toby that one should say “$(r+1,s)$-order” (I don’t use “poset”, since this word suggests that they have a set of objects) rather than “$(n,r)$-category” and even $V$-order instead of $V$-category, because the word “category” carries the further meaning that there is some groupoidality (the homs are groupoidal orders in a category). For this reason, I think that “order” conveys better the general behaviour than “category”. The most fundamental concepts, for me, are the $n$-orders, which are the most general $\omega$-orders of dimension $n$. The study of higher-dimensional toposes would perhaps be more natural if the reference cases were modelled on Ord and 2-Ord rather than Set and Cat. —Mathieu

I disagree that “category” carries the meaning of groupoidality in the homs. I suppose it can mean whatever you choose it to mean, like Humpty Dumpty, but there is nothing in the past usage of the word “category” to suggest such an implication. Just because $n$-categories, for natural numbers $n$, happen to have homs that are groupoidal at the top dimension for which they have unequal parallel morphisms doesn’t mean that “category” necessarily implies such groupoidality. If in addition to 2-categories we allow (2,1)-categories then (2,3)-categories are just as natural. The existing meaning of $V$-category includes categories enriched over orders (= (1,2)-categories) and categories enriched over truth values (= (0,1)-categories); why not allow “category” its existing, more inclusive, meaning?

Also, even in an $n$-order (= $(n-1,n)$-category) there is some groupoidality; it just doesn’t kick in until the $(n+1)$-cells. To me, a much stronger intuitive association is that “order” implies that parallel morphisms are equal. This is not true for $n$-orders for any $n\gt 1$, except at the $n$-cells, and is not true for any dimension of cells in an $\omega$-order (= $\omega$-category). -Mike

My use of the phrase ‘fundamental concept’ was tongue in cheek, since (as you point out) it is the same concept. At best, one can say that the partial function from $\mathbf{Z} \times \mathbf{Z}$ to properties of $\infty$-categories that maps $(s,r)$ to being an $(s,r)$-poset (or $(s,r)$-order) is simpler (hence should be regarded as more fundamental, with the other defined in terms of it) than the partial function that maps $(s,r)$ to being an $(s,r)$-category. Certainly the domain of the first partial function is nicer. So if we were inventing terminology afresh for the whole business, we would probably want to base it on that function. Even in the real world, where I titled an article (n,r)-category, one should talk about $(s,r)$-posets when that helps explain a pattern.

Also, we're not going to get anywhere talking about the limiting connotations that various words have. Of course ‘category’ has the connotation of being groupoidal at and above level $2$, since an ordinary category is groupoidal at and above level $2$; and ‘order’ and ‘poset’ have the connotation of being trivial at and above level $2$, since an ordinary order or poset is trivial at and above level $2$. (But I don't see what's wrong with the connotation that a poset has an underlying set, since even an $\infty$-poset has an underlying set of equivalence classes of objects.) Any terminology that we choose to generalise will have limiting connotations from its previous use.

I think this is the last I'll say on the matter; it's not like I'm not trying to get Mike to renumber his work here.

—Toby