The n-Category Café

I worked a lot today and feel tired now. In order to relax a bit I opened the above lecture notes and learned about (-1)-categories and (-2)-categories. (I had wanted to do this a while ago already.)

An $n$-category is a $\mathrm{Hom}$-thing of an $(n+1)$-category.

Hence a (-1)-category is a $\mathrm{Hom}$-thing of a 0-category.

And a (-2)-category is a $\mathrm{Hom}$-thing of a -1-category.

We are being asked to figure out what a monoidal (-1)-categories is.

That’s not hard. At least not if you feel sufficiently relaxed about the meaning of the word “is”.

For me, the really hard part is to figure out what the difference between a monoidal (-1)-category and a (-2)-category is.

They shouldn’t be exactly the same, should they?

Hey, this blog is also about philosophy. So I am still on topic!!

As John discusses, calling the two possible (-1)-categories “true” and “false” or “equal” and “not equal” both seem to be reasonable choices.

Had I been asked for a name, though, I would probably have suggested “discrete category on 1-object set” and “discrete category on empty set”, instead.

After all, if we regard a 0-category $S$ as an $\infty$-category with all $p$-morphisms identities for all $p$, then $\mathrm{Hom}_S(x,x)$ is the $\infty$-category with a single object (namey the identity 1-morphism on $x$) and all $p$-morphisms identites. So it’s still an infinity-category!

Similarly $\mathrm{Hom}_S(x,y)$ for $x \neq y$. This is the $\infty$-category with no object.

That’s what my answer would have been.

However, with that answer I would have found that the unique (-2)-category is also an $\infty$-category with a single object and all $p$-morphisms identities.

Same for monoidal (-1)-category. This is a 0-category with a single object, hence once again an $\infty$-category with a single object and all morphisms identies.

So it seems I am in trouble. I cannot distinguish between the unique monoidal 0-category, the unique (-2)-category and one of the two (-1)-categories.

However, from p.14 of the lecture notes mentioned above one finds that the experts distinguish the non-empty (-1)-category from the unique (-2)-category by using the distinction between TRUE and NECESSARILY TRUE.

Heh. It’s true that $e^{\pi i} = -1$.

Or so I thought.

Maybe instead it is necessarily true??

How can I tell?

Is there a (-1)-category of solutions to the equation $e^{\pi i} = -1$, or a (-2)-category. Or a monoidal (-1)-category?

OK, I go to bed now. Seems about time.

But before quitting, one more serious comment.

We use 1-functors to describe charged particles.

We use 2-functors to describe charged strings.

We use 3-functors to describe charged membranes.

We use 0-functors to describe charged instantons, aka $(-1)$-branes with a 0-dimensional worldvolume.

Really, we do. At least some people do ($\to$).

Do we need (-1)-functors for anything?

Let’s see. By my way of looking at this situation, a $(-1)$-functor is the same as a 0-functor on a monoidal 0-category.

OK, I can interpret that. That’s a charged instanton which is constrained to appear at a predescribed point in spacetime.

Not any point. A fixed point. The point being the single object in our monoidal 0-category.

All right, now it’s official. A $D(-2)$-brane is a D-instanton with predescribed point of appearance.

Maybe I should say it’s a D-instanton living in a 0-dimensional target space.

Yeah, that sounds right.

I was looking again at the Lectures on $n$-Categories and Cohomology to see if could find a hint towards the answer of the following question:

What’s an action $n$-groupoid and how is it characterized by morphisms to an $n$-group?

In week 249 John mentions that

Any groupoid with a faithful functor to $G$ is equivalent to the action groupoid $X//G$ for some action of $G$ on some set $X$.

I like to think of it this way: I write $\mathbf{B} G$ for the group $G$ regarded as a one-object groupoid and then notice that every action groupoid sits in a sequence of groupoids

$$X \to X//G \to \mathbf{B} G \,,$$

where the $X := Disc(X)$ on the left is the discrete groupoid over $X$ ($X$ as objects, no nontrivial morphisms).

In particular, if $X=G$ and we have the right action of $G$ on itself we get

$$G \to G//G \to \mathbf{B} G \,,$$

which is the groupoid version of the universal $G$-bundle, realizing $\mathbf{E}G := G//G$ as the action groupoid of $G$ action on itself.

What I want to understand is: How does this generalize to higher $n$?

Given an $n$-group $G$ and writing $\mathbf{B} G$ for its one-object $n$-groupoid incarnation, which sequences of $n$-groupoids

$$X \to X//G \to \mathbf{B}G$$

I am entitled to address as coming from action $n$-groupoids?

I was looking at the lecture notes by John and Mike since they talk about the generalization of “faithful” to higher $n$ (e.g. p. 18).

There the observation of course is that for 1-functors, faithful means surjective on 2-morphisms (identities).

So I suppose it would make sense to call an $n$-functor faithful if it is $(n+1)$-surjective, i.e. surjective on equations between $n$-morphisms. But does that lead to the right characterization of action $n$-groupoids?

It looks intuitively all right, but one problem is that one would hope there to be a notion of $\infty$-action groupoid, but what’s $(\infty+1)$-surjective?

In principle it should be straightforward to work it out:

for $G$ an $n$-group and $C$ an $n$-category with a forgetful (hm…) $n$-functor to $(n-1)Grpd$, we should look at actions of $G$ on objects in $C$ $$\rho : \mathbf{B} G \to C$$ then send this forward to $n Grpd$

$$\hat \rho : \mathbf{B} G \stackrel{\rho}{\to} C \to (n-1)Grpd \hookrightarrow n Grpd$$

and then take a weak colimit there:

$$X//G := colim_{\mathbf{B} G}\rho \,.$$

That’s how it works for ordinary action 1-groupoids.

By the universal property of the colimit we canonically get the morphism

$$X//G \to \mathbf{B} G$$

and one should check what characteizes the morphisms obtained this way. Such that one could say that for every $n$-functor

$$V \to \mathbf{B} G$$

with these and those properties, $V$ is equivalent to a colimit of the above kind.

Does anyone know what these properties are? It depends on what we take a “weak $n$-colimit” to be.

I’d be most happy if we could do this in $\omega Cat$ (strict $\infty$-categories). But whatever works, I’d be glad to see it.