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Stream: theory: algebraic topology

Topic: describing simplicial sets

Leopold Schlicht (Nov 05 2021 at 14:50):

Is there some way of describing a simplicial sets by just giving the face maps (and not the degeneracy maps)?

Let me give you two motivations and one guess in the direction of a possible answer.

In a previous thread, I gave the following naive definition of $\Delta^2$ and asked whether it can be considered to be equivalent to the official definition of $\Delta^2$: we have three 0-simplices, three 1-simplices, and one 2-simplex; no $n$-simplices for $n>2$. The face maps are given in the evident way. So I tried to define $\Delta^2$ by just giving the face maps and just giving the nondegenerate simplices.

I stumbled across another class of simplicial sets I wished I could specify without giving the face maps and the degenerate simplices: directed graphs. From each simplicial set of dimension $\leq 1$ one can build a directed graph and vice versa. With a suitable definition of "morphism between directed graph" these constructions even induce an equivalence between the category of simplicial sets $\leq 1$ and the category of directed graphs. Now, a directed graph is given by two sets, $V$ and $E$, and two arrows $E\to V$. These data can be considered as providing the nondegenerate simplices of a simplicial set with dimension $\leq 1$: $V$ is the set of 0-simplices, $E$ the set of nondegenerate 1-simplices, and the two arrows $E\to V$ provide the two face maps from 1-simplices to 0-simplices.

Is there some easy way of extracting the degeneracy maps and the degenerate simplices in all dimensions from that information? How to describe the resulting simplicial sets concretely?

In the above link, the simplicial set induced by a directed graph is described as a pushout: just take $|E|$ copies of $\Delta^1$, i.e., $|E|$ arrows and glue them, in a way specified by $s$ and $t$, onto $|V|$ copies of $\Delta^0$, i.e., $|V|$ points (see the pushout diagram in the proof). This is nice and of course uniquely specifies the resulting simplicial set, but I want to see a concrete description of the resulting simplicial set in terms of sets of $n$-simplices and face and degeneracy maps. And I'm curious whether one can come up with a general formalism that allows one to specify a simplicial set by just giving the nondegenerate simplices and the face maps. (In particular, that formalism should yield the real definition of $\Delta^2$ if I put in my naive definition of $\Delta^2$.)

I suspect that the construction I am searching for should have the following universal property: let $\Delta'$ be the subcategory of the simplex category $\Delta$ consisting of only the strictly decreasing maps (but all objects of $\Delta$). Then a semisimplicial set is a presheaf on $\Delta'$. There is a forgetful functor from simplicial sets to semisimplicial sets. Probably the construction I am searching for is a left adjoint of that functor.

How to construct that left adjoint? Also, which simplicial sets are free in that sense? The above discussion suggests that all simplicial sets of dimension $\leq 1$ and all standard simplices $\Delta^n$ are free. What would be an example of a simplicial set that isn't free, that is, a simplicial set that can't be described with just face maps and nondegenerate simplices?

Reid Barton (Nov 05 2021 at 16:01):

There's a phenomenon in the general case that you don't see in the low dimension of directed graphs: a face of a nondegenerate simplex can be degenerate. For example, you could take the quotient of $\Delta^2$ by one of its edges (or by its entire boundary).

Leopold Schlicht (Nov 05 2021 at 17:41):

How is the quotient of a simplicial set by an edge defined? So are you claiming that this particular simplicial set isn't free? (Thanks for providing that example! :smiley:)

What's your take on the other questions and thoughts I wrote down?

John Baez (Nov 05 2021 at 19:00):

I imagine to "take the quotient of $\Delta^2$ by one of its edges" you just decree that edge to be the degeneracy of both its vertices, and impose all the equations that follow from this via the simplicial identities.

John Baez (Nov 05 2021 at 19:01):

You can impose any set of equations you want between simplices in a simplicial set, as long as you also impose all the equations that follow via the simplicial identities. You'll get a new simplicial set this way, which will be a quotient of the original one.

John Baez (Nov 05 2021 at 19:04):

If we impose an equation saying an edge in a triangle is the degeneracy of both its vertices, the simplicial identities force these two vertices to be equal. (They also force other things to be equal.)

John Baez (Nov 05 2021 at 19:04):

Basically I'm just trying to take the edge and "collapse it down to a point".

John Baez (Nov 05 2021 at 19:06):

Perhaps I'm being redundant and we only need to demand that the edge is the degeneracy of one of its vertices. But whatever... I'm just trying to get the job done.

Daniel Teixeira (Nov 07 2021 at 21:01):

Leopold Schlicht said:

How to construct that left adjoint?

From a semi-simplicial set $X$ we construct a simplicial set $\tilde X$ that's obtained by "freely adding degeneracies".

First let $\tilde X_0 = X_0$. For $n = 1$, let $X_1\subseteq \tilde X_1$, also for each $x\in \tilde X_0$ and $0\leq i \leq 1$ there is an element $s_i(x) \in\tilde X_1$ to serve as the i-th degeneracy. Now we proceed inductively: we let $X_{n+1}\subseteq \tilde X_{n+1}$, also for each $x\in \tilde X_n$ and $0\leq i \leq n+1$ there is an element $s_i(x) \in\tilde X_{n+1}$.

To make $\tilde X$ a genuine simplicial set, you have to quotient the sets $\tilde X_n$ by the simplicial identities. E.g. we freely added the (at this point different) elements $\tau = s_2\circ s_3(x)$ and $\sigma = s_4\circ s_2(x)$, so we identify $\tau = \sigma$.

Daniel Teixeira (Nov 07 2021 at 21:04):

I think this is right. Anyway, at nLab [[semi-simplicial set]] there is an abstract construction of both left and right adjoints for the forgetful functor you mention. Also, for any simplicial set $X$, the left unitor gives a weak htpy equivalence to the image of a semi-simplicial set through this adjunction.

Daniel Teixeira (Nov 07 2021 at 21:08):

You may also enjoy this proposition at Kerodon, and a corollary from it that says that a simplicial map that is a bijection on non-degenerate simplices is an isomorphism

Reid Barton (Nov 08 2021 at 15:48):

There's a lot more interesting stuff to say about this situation.

Reid Barton (Nov 08 2021 at 15:49):

If we want to understand the category of semisimplicial sets relative to the category of simplicial sets via the left adjoint you mentioned, we should answer three questions:

1. Which simplicial sets lie in the image of this functor?
2. Which morphisms lie in the image of this functor?
3. When do two parallel morphisms of semisimplicial sets become equal when we apply this functor?

Reid Barton (Nov 08 2021 at 15:50):

I think, though I haven't carefully checked, that the answers are:

1. The simplicial sets in which every face of a nondegenerate simplex is nondegenerate.
2. The morphisms which send nondegenerate simplices to nondegenerate simplices.
3. Only if the maps were already equal, i.e., the functor is faithful.

Reid Barton (Nov 08 2021 at 15:58):

There's also a more efficient way to describe what this left adjoint produces, related to the kerodon proposition that Daniel linked to, and using the notion of a "degeneracy operation". A degeneracy operation is an operation taking $n$-simplices to $m$-simplices for some fixed $n$ and $m$, for which the corresponding map $[m] \to [n]$ of $\Delta$ is surjective. (So in particular, $n \le m$.) The operations $s_i$ are the generating degeneracy operations, and the degeneracy options are all compositions of the $s_i$, but quotiented by the simplicial relations involving the $s_i$.

The linked proposition says that every simplex of a simplicial set can be expressed as a degeneracy operation applied to a nondegenerate simplex in a unique way.

Reid Barton (Nov 08 2021 at 16:05):

Now if we start with a semisimplicial set $X$, we can describe the "free" simplicial set $Y$ it generates as follows:

• The simplices of $Y$ are formal applications of a degeneracy operation to a simplex of $X$.
• The structure maps of $X$ are computed as follows. Suppose we want to compute the action of a simplicial operator $f$ on a formal degeneracy $sx$. The combined operation $fs$ corresponds to some map of $\Delta$ which we can refactor as a surjection followed by an injection. Then $f(sx)$ is given by formally applying the degeneracy operator corresponding to the surjection to the value of the face operator corresponding to the injection on $x$ (computed in the semisimplicial set $X$).

Reid Barton (Nov 08 2021 at 16:10):

A more syntactic way to describe the action in terms of the generating face and degenerating operators is:

• If we want to apply $s_i$ to a formal degeneracy $sx$, we just form $(s_i s) x$.
• If we want to apply $d_i$ to a formal degeneracy $sx$, then we use the simplicial identities to rewrite $d_i s$ as a composition $s' d'$ by "moving ds to the left". Since we started with a single $d_i$, what will happen is that either $d_i$ will pass through all the ss (possibly changing indices in the process) so that $d = d_j$, or the $d_i$ will cancel with some s, so that $d = \mathrm{id}$. Then we compute $x' = d' x$ in $X$ and form $s' x'$.

Reid Barton (Nov 08 2021 at 16:16):

Finally, there is also a way to specify an arbitrary simplicial set in terms of only its nondegenerate simplices and its face maps, but with the caveat that the face of a nondegenerate simplex can be a formal degeneracy of another nondegenerate simplex. The full simplicial structure is recovered by the same process as above except that when we take the face of a nondegenerate simplex (in what would have been $X$ above), it may come as a formal degeneracy to which we have to apply another degeneracy operator to--which is no problem.

Reid Barton (Nov 08 2021 at 16:17):

The other caveat is that because of the original question 2, in order to recover the correct maps of simplicial sets, we also need to allow a map to send a nondegenerate simplex to a formal degeneracy in the target simplicial set.

Reid Barton (Nov 08 2021 at 16:18):

The program Kenzo uses this representation of simplicial sets.

Leopold Schlicht (Nov 13 2021 at 14:34):

@Reid Barton @Daniel Teixeira Thanks a lot, that really helps a lot!

Patrick Nicodemus (Nov 17 2021 at 13:17):

Reid Barton said:

The program Kenzo uses this representation of simplicial sets.

Whoa. Never heard of this. super cool.

Leopold Schlicht (Nov 18 2021 at 18:16):

John Baez said:

You can impose any set of equations you want between simplices in a simplicial set, as long as you also impose all the equations that follow via the simplicial identities. You'll get a new simplicial set this way, which will be a quotient of the original one.

What do you mean by that? What confuses me is that when quotienting a simplicial set by an internal equivalence relation the simplicial identities aren't involved at all. (I think one rather has to check that the face and degeneracy maps preserve the relation one wants to quotient by.)

Reid Barton (Nov 18 2021 at 19:27):

Leopold Schlicht said:

John Baez said:

You can impose any set of equations you want between simplices in a simplicial set, as long as you also impose all the equations that follow via the simplicial identities.

(I think one rather has to check that the face and degeneracy maps preserve the relation one wants to quotient by.)

These seem like two different ways of saying the same thing

Leopold Schlicht (Nov 18 2021 at 20:11):

To me not. In my way of saying it the simplicial identities aren't involved.

Reid Barton (Nov 18 2021 at 20:44):

If your relation is already compatible with face/degeneracy maps, then you don't have to "also" impose any relations that follow from their action (because they are already in the original relation)

John Baez (Nov 19 2021 at 00:07):

Of course that's correct, Reid. And on the other hand, you can mod out by imposing any equations you like, without "checking" any properties of these equations, as long as you also impose all equations that follow from these using the definition of simplicial set.

John Baez (Nov 19 2021 at 00:12):

To take a similar example, I can take a group containing two elements $x$ and $y$ and impose the equation $x = y$, and also all equations that follow from this via the definition of group (like $xz = yz$, etc.), and I'll get a new group, which is a quotient of the original one.

It works like this for any algebraic gadget defined by equations, including multisorted gadgets like simplicial sets. I could state this more formally, but it's easy enough to see it's true.

John Baez (Nov 19 2021 at 00:18):

But I didn't say it correctly when I said "follow via the simplicial identities". I should have said "follow via the definition of simplicial set". In an algebraic gadget, we get new equations from old ones using operations as well as from equations. In the case of a simplicial set, these operations are the face and degeneracy maps.

John Baez (Nov 19 2021 at 00:35):

So, if we identify two simplexes (of the same dimension), we have to also identify their corresponding faces, and their corresponding degeneracies.

Leopold Schlicht (Nov 23 2021 at 15:26):

I found out that the adjunction discussed in this thread is often used implicitly in Kerodon, so thanks again for all the answers.