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Stream: learning: questions

Topic: simplicial sets

Morgan Rogers (he/him) (Jul 01 2020 at 15:46):

I'm currently looking at subtoposes of the topos $[\Delta^{\mathrm{op}},\mathbf{Set}]$ of simplicial sets. Can someone give me a good reference for the n-skeleton/n-coskeleton comonad/monad pair on this category?
n-truncation is a very intuitive construction on simplicial sets, but the way in which the adjoint functors extend a truncated complex back to a full complex involves some fiddly combinatorics that I want to get a better handle on.

Morgan Rogers (he/him) (Jul 04 2020 at 14:50):

That is to say, what do skeletal and coskeletal simplicial sets look like? If I have a simplicial set $X$, what is its $n$-coskeleton? I know that the first $n$ terms are just the first $n$ terms of $X$, but what does this vague description on the nLab,

"The n-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree n. Conversely, the n-coskeleton produces a simplicial set having a simplice of degree m>n whenever there is a compatible family of m-faces,"

concretely mean?

John Baez (Jul 04 2020 at 15:04):

The n-coskeleton of X looks like X up to dimension n, and then you put in one (n+1)-simplex in every place you possibly can, and then you put in one (n+1)-simplex in every place you possibly can, and so on.

John Baez (Jul 04 2020 at 15:06):

For example if X is a hollow triangle (just three nondegenerate 1-simplices), its 2-coskeleton would be a triangle: you fill in the hole.

John Baez (Jul 04 2020 at 15:07):

There's no room for any nondegenerate 3-simplices, etc.

John Baez (Jul 04 2020 at 15:08):

So in terms of homotopy theory you can see we've killed off the $\pi_1$ and all higher homotopy groups (which in this particular case were already trivial, so there's nothing to do).

John Baez (Jul 04 2020 at 15:09):

In short: we fill all holes above a certain dimension.

Morgan Rogers (he/him) (Jul 04 2020 at 15:44):

Rongmin Lu said:

I think the Wikipedia page for n-skeleton seems to be more informative.

Is it? It tells me what the 0 skeleton and 0-coskeleton are, but no higher, and contains the the vague description

The idea of the n-skeleton is to first discard the sets $K_{i}$ with $i > n$ and then to complete the collection of the $K_{i}$ with $i\leq n$ to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees $i > n$.

How/why/when do we need to add anything to $K_{i}$ with $i\leq n$ in order to obtain the n-skeleton?

Morgan Rogers (he/him) (Jul 04 2020 at 15:47):

John Baez said:

The n-coskeleton of X looks like X up to dimension n, and then you put in one (n+1)-simplex in every place you possibly can, and then you put in one (n+1)-simplex in every place you possibly can, and so on.

Is there a typo here? Given that in the triangle example you said its 2-coskeleton is obtained by adding a 2-simplex (rather than a 3-simplex)

Morgan Rogers (he/him) (Jul 04 2020 at 15:51):

But in that case the 2-coskeleton of a triangle would be the triangle, not filled in

Morgan Rogers (he/him) (Jul 04 2020 at 15:55):

Rongmin Lu said:

There's this:

For example:
$\textrm{skel}_0$(cube) = 8 vertices
$\textrm{skel}_1$(cube) = 8 vertices, 12 edges
$\textrm{skel}_2$(cube) = 8 vertices, 12 edges, 6 square faces

I understood that as conflating the n-skeleton with the n-truncation, since a simplicial set can't have simplices in dimensions 0 and 1 without having at least one (presumably "degenerate", which is a word that keeps appearing but which I only have some idea of the meaning of from the topological examples) in each higher dimension.

Morgan Rogers (he/him) (Jul 04 2020 at 15:57):

(and the above only works in reference to the generators in the chain complex of abelian groups associated to the simplicial complex for the cube, right?)

John Baez (Jul 04 2020 at 16:04):

[Mod] Morgan Rogers said:

John Baez said:

The n-coskeleton of X looks like X up to dimension n, and then you put in one (n+1)-simplex in every place you possibly can, and then you put in one (n+1)-simplex in every place you possibly can, and so on.

Is there a typo here? Given that in the triangle example you said its 2-coskeleton is obtained by adding a 2-simplex (rather than a 3-simplex)

Yeah, sorry - I'm too lazy to sort out the fencepost errors. Starting in some dimension n, you throw in one simplex of that dimension in each place you can, then move up to the next dimension and throw in one simplex in each place you can, etc. I don't really care if this is called n-coskeletal or (n+1)-coskeletal or (n-1)-coskeletal.

John Baez (Jul 04 2020 at 16:05):

Of course if I were proving theorems using this idea I'd make sure my conventions matched the usual one, but I'm just lying in bed waking up sipping coffee here.

John Baez (Jul 04 2020 at 16:18):

Morgan wrote:

(presumably "degenerate", which is a word that keeps appearing but which I only have some idea of the meaning of from the topological examples)

Intuitively, degenerate simplices are "invisible" simplices. For example if I take any edge - that is, any 1-simplex in a simplicial set - with vertices A and B, there will be degenerate 2-simplices with vertices AAB and ABB. These are triangles that are so long and skinny that to the naked eye they look just like the edge you started with.

John Baez (Jul 04 2020 at 16:22):

Degenerate simplices seem so boring that originally people often left them out and worked with "semisimplicial sets" where you have face maps but no degeneracies. However, they're important!

For example, suppose you take the product of two copies of the "simplicial 1-simplex": the simplicial set freely generated by a 1-simplex: it looks just like a 1-simplex. (It's a representable presheaf.)

What do you get? You get a square subdivided into 2 triangles! Where do these triangles come from? They come from taking products of degenerate 2-simplices! They are not themselves degenerate.

This is a crucial calculation to do, to understand simplicial sets.

Morgan Rogers (he/him) (Jul 04 2020 at 16:29):

Thanks! So the degenerate ones are the ones that you can ignore from the perspective of geometric realisations of simplicial sets, but they come into play in a crucial way when you're trying to perform operations on simplicial sets.

John Baez (Jul 04 2020 at 16:30):

Right!

John Baez (Jul 04 2020 at 16:30):

The "crucial calculation" is what I needed to do to understand why geometric realization preserves products.

John Baez (Jul 04 2020 at 16:31):

Since an n-simplex in a product of simplicial sets is just a pair of n-simplices in the two simplical sets, at first it seems impossible for geometric realization to preserve products.

John Baez (Jul 04 2020 at 16:32):

How are you going to get 2-simplices when you take the product of two 1-dimensional things?

John Baez (Jul 04 2020 at 16:33):

The answer is that any simplicial set has lots of high-dimensional simplices hidden in it - degenerate ones - that are waiting to blossom forth into nondegenerate ones when you take products of them.

Matt Feller (Jul 05 2020 at 19:03):

The way I think about $n$-coskeletal simplicial sets is via the "unique boundary extension" condition, which is mentioned on the nlab page and is what @John Baez was getting at by talking about filling in all possible holes above a certain dimension. More explicitly, this condition says that for $k>n$, every $k$-simplex is uniquely determined by its boundary. It means that all those $k>n$ simplices are pretty much just there to tell us that the lower dimensional simplicies fit together in a certain way.

I think a good example to understand is the nerve of a category, which is 2-coskeletal, but not necessarily 1-coskeletal. Given a triangle of 1-simplices in the nerve, i.e., a choice of $f\colon x\to y$, $g\colon y\to z$, and $h\colon x\to z$, in our category, the existence of a 2-simplex filler tells us something which the boundary alone didn't: that $g\circ f=h$. So, the nerve of a category is (usually) not 1-coskeletal because the 2-simplices are giving us extra information which wasn't contained in the 1-simplices. Now, a hollow tetrahedron of 2-simplices in the nerve amounts to a choice of three composable morphisms in our category along with all of the different ways we could compose them. But that data is precisely the information of a 3-simplex! So nerves of categories are 2-coskeletal, because the information of a 3-simplex (or higher) is contained entirely in its boundary.

Morgan Rogers (he/him) (Jul 06 2020 at 08:47):

Matt Feller said:

I think a good example to understand is the nerve of a category, which is 2-coskeletal, but not necessarily 1-coskeletal.

Thank you very much for this example. It illustrates that an assumption I was implicitly making is false: viewing the (tr ${\dashv}$ cosk) adjunction as geometric inclusions, making the $n$-truncation toposes into subtoposes of the category of simplicial sets, I had assumed that these subtoposes would fit together into a chain of inclusions. Apparently they do not! Since I am considering another subtopos, this makes the question of how it intersects with the $n$-coskeleton subtoposes a lot more interesting.

Reid Barton (Jul 06 2020 at 11:18):

Don't they? A 1-coskeletal simplicial set is also 2-coskeletal, and so on.

Morgan Rogers (he/him) (Jul 06 2020 at 11:24):

:face_palm: looks like I got the arrows the wrong way around, thanks Reid.