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Stream: deprecated: id my structure

Topic: some categories of shapes

Mike Shulman (Jan 22 2023 at 17:55):

Let $S_k$ be the strict monoidal category freely generated by an object $G$ and $k$ morphisms $I\to G$, where $I$ is the monoidal unit. What is a good name for presheaves on $S_k$ for general $k$?

• When $k=1$, this is the category of augmented semisimplicial sets. For a presheaf $X$, we have $X(G^n) == X_{n-1}$, the set of $(n-1)$-simplices; thus $X(I) = X(G^0) = X_{n-1}$ is a set of $(-1)$-simplices.

• When $k=2$, this is the category of semicubical sets, where $X(G^n) = X_n$ is the set of $n$-cubes. Note the interesting dimension-shift.

I'd also be interested in names for the analogues where the category is symmetric or semicartesian monoidal, although the former could presumably be obtained by adding the adjective "symmetric", and the latter by removing the prefix "semi-" if it's present in the general name.

Mike Shulman (Jan 22 2023 at 17:56):

In at least one place I think I've seen these called something like "$k$-ary semicubical sets", but I'm not really happy with that, since they don't seem very cubical at all for $k\neq 2$.

James Deikun (Jan 22 2023 at 18:34):

They're going to be more like cubes than like simplices for $k > 1$ because they'll retain the characteristic of $G^n$ having exponentially many points. Similar to how unary numbering is vastly different from $k$-ary for $k > 1$. They're certainly not going to turn into associahedra or anything.

James Deikun (Jan 22 2023 at 18:37):

Specifically, they'll be like cubes where each edge has $k$ points on it, but in no particular order.

James Deikun (Jan 22 2023 at 18:44):

If you really want to avoid suggesting they're cubical, maybe "semi-$k$-furcated sets" though.

Mike Shulman (Jan 22 2023 at 20:19):

I can't see any way in which they are like cubes for $k>2$. Yes, for $k=3$ a 1-thing has three 0-things "on" it, but a 2-thing has nine 1-things on it. What sort of a square has nine edges?

Mike Shulman (Jan 22 2023 at 20:20):

There are lots of sequences of polytopes with exponentially many vertices.

Morgan Rogers (he/him) (Jan 23 2023 at 07:16):

How did you get 9 1-things? I count 6

Morgan Rogers (he/him) (Jan 23 2023 at 07:17):

(assuming these are morphisms $G \to G,G$)

Amar Hadzihasanovic (Jan 23 2023 at 08:21):

I guess one way of seeing $S_k$ is as the monoidal subcategory of the augmented simplex category generated by the facets (codimension-1 faces) of the $(k-1)$-simplex. So the augmented simplex category is itself generated under joins by the only facet of the 0-simplex, $S_2$ by the two facets of the 1-simplex (and happens to be isomorphic to semicubical sets), etc.

The “grouping” when pictured in string diagrams for the augmented simplex category makes me think of tuplets in music notation, so perhaps I'd think of calling it the “$k$-tuplet semisimplicial sets” or something like that.
image.png

Mike Shulman (Jan 23 2023 at 08:30):

@Morgan Rogers (he/him) You're right, sorry. Six 1-things and nine 0-things.

Mike Shulman (Jan 23 2023 at 08:33):

@Amar Hadzihasanovic That doesn't make sense to me. For one thing, a monoidal subcategory would always include the unit object. For another, I can't see how you are thinking of the semicube category as embedded in the semisimplex category.

Amar Hadzihasanovic (Jan 23 2023 at 08:48):

What I mean is: you “black-box” the $(k-1)$-simplex and $(k-2)$-simplex.
The objects of your monoidal subcategory are joins of $(k-1)$ and $(k-2)$-simplices, including the empty join which is the unit object.
The morphisms are generated under joins by the identities on the $(k-1)$ and $(k-2)$-simplices, and by the $k$-many face inclusions of the $(k-2)$-simplex into the $(k-1)$-simplex.

Amar Hadzihasanovic (Jan 23 2023 at 08:48):

Am I missing some reason why this wouldn't work?

Amar Hadzihasanovic (Jan 23 2023 at 09:02):

Ah, I see, the $(k-2)$-simplex needs to be the unit in $S_k$, oops

Amar Hadzihasanovic (Jan 23 2023 at 09:18):

So instead I guess you only have a functor from the monoidal subcategory that I described to $S_k$, which sends the $(k-2)$-simplex to the unit, but not a functor the other way around; although you would have a “canonical” way to lift morphisms of $S_k$, so perhaps this functor is at least some nice kind of fibration

James Deikun (Jan 23 2023 at 12:57):

Think of the "square" in the 3-case being like a window pane in a child's drawing of a house: a square with a cross in the middle. The choice of which edges are "middle" is arbitrary because there's no structure that distinguishes which points are the "corners", but you could embed this in a 2-category where the $k$ generating morphisms are ordered and then everything would turn out like this precisely: for $k+2$, $G^n$ would be an $n$-cube divided along each axis by $k$ internal hyperplanes. You could even see the case for $k = 1$ as being about cubes where you only care about the vicinity of one corner, and then you cut near that corner with a diagonal hyperplane to get the simplex view (and the cutting causes the "curious" dimension shift).

Mike Shulman (Jan 23 2023 at 17:00):

Oh! I think I see. Yes, I think with that intuition I can see them as cubes. Thanks.