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.

## Re: Lectures on n-Categories and Cohomology

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

differencebetween 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.