The n-Category Café

Incomplete as it is, maybe the $n$Lab entry on subobject ($\to$ $n$Lab) is already better than the Wikipedia entry.

There are nice concepts that generalise both abelian categories and toposes. But that will probably not work, since constructively $\Sub(c \setminus x)$ is only a retract of $[x,c]$.

Is there a nice concept that generalises both abelian categories and boolean toposes? (Incidentally, even constructivists believe that $\Fin\Set$, if appropriately defined, is boolean.)

I have just finished my PhD thesis and I am getting very interested in the categorification of various familiar mathematical structures. I am following n-cat cafe often and I wanted to say that I have found it very enjoyable.

Also, could someone tell me what the categorification of the symmetric group of n letters S_n is?

Maybe the following could help? http://archive.numdam.org/ARCHIVE/SG/SG_1957__1_/SG_1957__1__A1_0/SG_1957__1__A1_0.pdf

It is the first talk of the first “Séminaire A.Grothendieck”. Take a look at “3- Sous-trucs,trucs quotients” pp 1-03,1-04.

Both David and Toby made some suggestive comments in reply to John’s question; I’d like to put down some things they suggested to me, which I think give a solution to John’s problem.

I’ll interpret the problem as finding some exactness conditions on a category $C$ that ensure the existence of the interval “quotient” John is after. Here’s one set of possibilities: suppose $C$ is a category such that

1. $C$ is finitely complete and finitely cocomplete;
2. The pushout of any mono along any morphism is a mono, and such pushout squares are also pullbacks;
3. The pullback of any epi along any morphism is an epi, and such pullback squares are also pushouts.
4. For any object $x$, the arrow $0 \to x$ is monic; if $x$ is not initial, then the arrow $x \to 1$ is epic.

Notice that conditions 1, 2, 3 taken together are self-dual. Condition 4 is not, but it’s somewhat in that vein; it reads that way partly on account of the fundamental schism between toposes and abelian categories discussed by Freyd in the notion of AT category.

(It’s true that in a topos, every arrow $0 \to x$ is monic. Of course it’s not generally true that in a topos, every arrow $x \to 1$ is epic, even if $x$ is not initial. But then again, not every topos has the sorts of objects John is looking for either, so some sort of restriction is called for. More on this below.)

If $C$ satisfies conditions 1-4, then we can prove a result (proposition 1) which is sort of a halfway house on the road to the objects John wants. Let $c$ be an object, $x \leq c$ a subobject. Define $c/x$ to be the pushout of

$$1 \leftarrow x \hookrightarrow c$$

(NB: not the same in general as what John was calling $c/x$ – we’ll get to that later.)

In particular, $x/x$ is terminal, and since pushouts of monos are monos, every chain of subobjects

$$x \leq y \leq c$$

gives rise to a chain of subobjects

$$x/x \leq y/x \leq c/x$$

(notice $y/x$ and $c/x$ are pointed objects, where the point in each is $x/x$).

If $x \cong 0$, then $c/x$ just adds a point to $c$: $c/0 \cong c + 1$. However, we can ignore this case: since $0 \to x$ is assumed monic for any $x$, we have that $[0, c] = Sub(c)$ anyway, making John’s problem trivial. So assume $x$ is not initial.

Proposition 1: The interval $[x, c]$ is isomorphic to the interval $[x/x, c/x]$, via the assignment $y \mapsto y/x$ for $y \in [x, c]$.

Proof: We claim the inverse map is the one which takes $z \in [x/x, c/x]$ to the pullback of the monic $z \hookrightarrow c/x$ along the pushout projection $c \to c/x$. This is a monic (denoted $z^\sim \hookrightarrow c$) that contains the subobject $x \leq c$, so $z^\sim \in [x, c]$.

Start with $y \in [x, c]$. Then the pushout

$$\array{ y & \hookrightarrow & c \\ \downarrow & & \downarrow \\ y/x & \hookrightarrow & c/x, }$$

being the pushout of a monic, is also a pullback by condition 2, and hence $y = (y/x)^\sim$ (as elements of $[x, c]$).

Now take $z \in [x/x, c/x]$. Since we assume $x \cong 0$ is false, the arrow $x \to 1$ is epic (condition 4), and so $c \to c/x$ is epic as well (pushouts of epis are epi). Therefore the pullback of $c \to c/x$ along $z \hookrightarrow x$, which is the map $z^\sim \to z$, is epi as well (condition 3). We have now a pullback

$$\array{ x & \hookrightarrow & z^\sim \\ \downarrow & & \downarrow \\ x/x & \hookrightarrow & z }$$

which is also a pushout, by condition 3. It follows that $z = z^\sim/x$ (as elements of $[x/x, c/x]$). The proof is now complete. $\Box$

To get the objects John wants, assume one more condition (condition 5):

• Every point $p: 1 \to a$ has a complement $\neg p \leq a$.

Under conditions 1-4, this amounts to existence of a square

$$\array{ 0 & \to & \neg p \\ \downarrow & & \downarrow \\ 1 & \stackrel{p}{\to} & a }$$

which is simultaneously a pushout and a pullback (a “dolittle square”).

Of course, this condition is trivially satisfied whenever $0 \cong 1$ (e.g., abelian categories).

Proposition 2: The complement of $x/x \to c/x$, denoted $(c/x)^{\bullet}$, has the property that

$$Sub((c/x)^{\bullet}) \cong [x, c]$$

Proof: Since $0 \to y$ is monic for any $y$, it suffices to prove (by proposition 1) that

$$[0, (c/x)^{\bullet}] \cong [x/x, c/x] = [1, c/x]$$

Put $a = c/x$. The map $[0, a^\bullet] \to [1, a]$ takes $y \in [0, a^\bullet]$ to $1 + y \in [1, a]$; in the other direction, we take $z \in [1, a]$ to $z \wedge a^\bullet \in [0, a^\bullet]$.

The proof that $(1 + y) \wedge a^\bullet = y$ uses the fact that the pushout of a mono $y \leq a$ is a dolittle square, much as in the proof of proposition 1. In the other direction, to prove $1 + (z \wedge a^\bullet) = z$, we just need to show $z^\bullet = z \wedge a^\bullet$ in order to again get a dolittle square; this equation is easy and is left to the reader. $\Box$

Remarks: conditions 1-3 hold in any AT category, such as an abelian category or a topos. However, the result that John is after cannot be expected to hold in a general topos. For example, in the topos of sheaves over the space $[0, 1]$, for instance, considered as local homeomorphisms $X \to [0, 1]$, the terminal object is $t = [0, 1]$. If we take the subobject $x = (0, 1]$, then the interval $[x, t]$ consists of two elements. But there is no sheaf over $[0, 1]$ with just two subsheaves! So what John is asking for cannot work in cases like this.

Generally speaking, it seems to me that the existence of subobjects strictly between 0 and 1 poses some awkwardness for the kind of thing John wants to consider, so my condition 4 is a kind of “two-valuedness” condition which precludes that. In a topos, two-valuedness implies Boolean [do I hear Toby piping in here? I’m assuming the metalogic is classical, in case there’s any doubt], so condition 5 is not that much of a stretch really.

I’d have to check, but I’m guessing that all these conditions are satisfied in the categories of interest mentioned by others: sets and modules of course, but I’m also guessing pointed sets. I’m not completely sold on the aesthetics of these conditions, but they seem at least to be a reasonable first step toward something more satisfactory.