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Stream: learning: questions

Topic: higher spines and generating simplices

Tim Hosgood (Nov 12 2020 at 14:52):

Given the standard $p$-simplex $\Delta^p$, its spine is the union of the edges joining consecutive points (i.e. the edge between the vertices labelled $0$ and $1$, along with the edge between the vertices labelled $1$ and $2$, along with ...). We say that the constituent $1$-simplices are generating, since, if we label each one with a morphism (from some fixed category) so that adjacent ones can be composed, then we can uniquely label the rest of the $1$-simplices in such a way that all triangles commute: we just use the compositions. We can think of this a bit more slickly as taking some $p$-simplex in the nerve of the fixed category and "gluing it" along the spine.

What I'm interested in is the higher-dimensional analogue of this. For example, what are the generating $2$-simplices? My first guess was that they would be $\{0<1<2\}$ and $\{1<2<3\}$, but it seems like they are actually $\{0<1<2\}$ and $\{0<2<3\}$. I'm sure this must be written down somewhere, but there was nothing on the nLab page for the spine (or, at least, nothing that I could understand as answering this question :upside_down: )

Reid Barton (Nov 12 2020 at 15:25):

Higher-dimensional analogue meaning 2-categories (and not (2,1)-categories)?

Reid Barton (Nov 12 2020 at 15:27):

I think if you want an analogue of the spine it would be better to replace $\Delta$ by something else, for example, $\Theta_2$.
If you want to work with $\Delta$, at a minimum, you'd have to specify how you want to regard simplices as representing 2-categories. But I think the answer will be that there isn't really a spine then.

Tim Hosgood (Nov 12 2020 at 15:34):

By "higher dimensional" I think I really just mean some specific union of $k$-simplices for $k>1$. I'm not really too sure what exactly I'm asking for, but my motivating example is really the one that I stated above: if we label faces of dimension $k$ by $k$-morphisms, then which faces do we need to label to ensure that the rest can be uniquely filled in by using compositions?

Tim Hosgood (Nov 12 2020 at 15:36):

so the diagram on the nLab page for the nerve seems to imply (I think) that what I want to call "the $2$-spine of $\Delta^3$" should be the union of the two faces $\{0<1<2\}$ and $\{0<2<3\}$ Screenshot-2020-11-12-at-15.35.34-1.png

Tim Hosgood (Nov 12 2020 at 15:41):

so I know what the ($1$-)spine of any $\Delta^p$ is, and I can just about understand what I want the $2$-spine of $\Delta^3$ to be, but I struggle to visualise anything higher than that I'm afraid...

Reid Barton (Nov 12 2020 at 16:13):

Are the triangles equations between parallel morphisms of a 1-category, or isomorphisms between parallel morphisms of a (2,1)-category, or general noninvertible 2-morphisms of a 2-category?

Reid Barton (Nov 12 2020 at 16:14):

oh I didn't realize that this diagram is from the nlab page

Reid Barton (Nov 12 2020 at 16:20):

I don't really understand what this diagram is supposed to convey, but let's say that we now have a 2-category and we imagine filling in each triangle with a 2-morphism that points down. Then it's not the case that the diagram on the left determines the diagram on the right because we have to choose a new decomposition of the square into two triangles.

Reid Barton (Nov 12 2020 at 16:23):

But first you had to choose a way to turn a simplex into a 2-category, in order to form the nerve at all.

Reid Barton (Nov 12 2020 at 16:23):

If you use the Duskin nerve, you'll see as pictured here that the 3-simplices again involve an equation between two different compositions of two 2-cells.

Reid Barton (Nov 12 2020 at 16:25):

So there's no way to label some of them so that the rest can be uniquely filled in (other than labeling all of them, I guess, if you want to allow that).

Tim Hosgood (Nov 12 2020 at 16:42):

I'm getting a bit confused about things I thought I understood now! (no fault except mine for that though).

Just to check: is my intuition for the ($1$-)spine correct? That it's exactly the collection of $1$-simplices of $\Delta^p$ such that, if we label them with a sequence of composable morphisms, then there is a unique way of labelling all the remaining $1$-simplices with morphisms so that every triangle commutes?

Reid Barton (Nov 12 2020 at 16:55):

Yes, that's right

Reid Barton (Nov 12 2020 at 16:58):

Simplices have a special relation to 1-categories because an object of $\Delta$ is a totally ordered set, which (being a poset) can also be viewed as a category.

Reid Barton (Nov 12 2020 at 16:58):

If you want an analogous story for 2-categories then you should probably replace $\Delta$ by something "2-dimensional", even though it is also possible to encode a 2-category by a simplicial set.

Joachim Kock (Nov 21 2020 at 21:54):

I don't know if this is of any use here, but I take the opportunity to mention it:

There is a different direction of generalisation of the spine condition: where the ordinary spine condition (essentially the Segal condition) encodes composition, there is something that in a sense encodes decomposition: these are called 2-Segal spaces (in the terminology of Dyckerhoff-Kapranov) or decomposition spaces (in the terminology of Gálvez-Kock-Tonks).

There are a few ways to describe the condition: DK describe it in terms of triangulations of plane polygons, in analogy with how the 1-Segal condition is about a kind of 'triangulation' of an interval. GKT describe it in terms of sendig certain pushouts in $\Delta$ to pullbacks, in analogy with writing the 1-Segal condition as $X_2 \simeq X_1 \times_{X_0} X_1$.