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I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my definitions are not precisely correct:

The category of simplicial sets is a cofibrantly generated model category with generating acyclic cofibrations the inclusions of all horns into standard simplicial sets. The map $\emptyset \to \Delta^0$ is such a horn inclusion, so it should be an acyclic cofibration.

However, an acyclic cofibration is a weak equivalence, so it should induce isomorphisms on all homotopy groups after realization. The realization of this map is the map $\emptyset \to \ast$ of topological spaces and this should be a weak equivalence, i.e. induce isomorphisms on all homotopy groups ($k \geq 0$) for all basepoints: $\pi_k (\emptyset,x) \to \pi_k(\ast,y)$

However, $\emptyset$ does not have any basepoints, such that the condition is empty, so that $\emptyset \to \ast$ is a weak equivalence. But this feels terribly wrong to me, since it implies that the emptyset is weakly homotopy equivalent to any space. A solution can be found by making a basepoint-independent definition of $\pi_0$ and set it to be equivalence classes of points. Then $\pi_0 (\emptyset) = \emptyset$ and $\pi_0(\ast) = \ast$, so $\emptyset \to \ast$ is not a weak equivalence.

But this would in turn imply that $\emptyset \to \Delta^0$ is not an acyclic cofibration of simplicial sets, because it is not a weak equivalence after realization.

So I do not find the right definition of $\pi_0$ to make it compatible with the Quillen adjunction, but I think it is just a mistake in my reasoning somewhere.

Any help would be greatly appreciated.

Alex

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Does make sense to compute homotopy groups for empty space? –  Sigur Nov 11 '12 at 0:28
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Certainly $\emptyset \rightarrow \Delta^0$ shouldn't be an acyclic cofibration, in any case. Probably you should just change your definition of horns. –  Aaron Mazel-Gee Nov 11 '12 at 0:53
    
I tried to find authors that actually specify which horns they are using, and right now I found two that did not include the map $\emptyset \to \Delta^0$ (they others did not say anything), so that is probably the problem. If we exclude this horn, everything will work out fine. –  Alexander Körschgen Nov 11 '12 at 2:37
    
However, this would contradict your comment in this thread math.stackexchange.com/questions/158162/… -- so Kan fibrations would not be necessarily surjective -- a counter example being any map starting from the empty simplicial set. –  Alexander Körschgen Nov 11 '12 at 2:39
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@Sigur: No, if you want to be precise then you should only be computing the homotopy groups of based spaces. Or you can pass to groupoids. –  Aaron Mazel-Gee Nov 11 '12 at 13:22

2 Answers 2

Thank you for your responses. Based on the discussion, the following conventions should be consistent and convenient:

  • The set of horn inclusions is the set $\{ \Lambda^n_k \to \Delta^n\ |\ n \geq 1,\ 0 \leq k \leq n \}$. This is the set of generating acyclic cofibrations for the standard model structure on (unpointed) simplicial sets.

  • A map $f: X \to Y$ of spaces is a weak equivalence, if $\pi_1 (X) \to \pi_1 (Y)$ is an equivalence of groupoids and $\pi_k (X,x) \to \pi_k (Y,f(x))$ is an isomorphism of abelian groups for $k > 1$.

Since we have $\pi_1 (\emptyset) = \emptyset$ and this groupoid is not equivalent to any other groupoid (which would arise from a non-empty space), this gives the empty set a distinct homotopy class.

This is consistent in the sense that it makes geometric realization into a left Quillen functor between simplicial sets and topological spaces (both unpointed).

For Kan fibrations, this implies that a Kan fibration is not necessarily surjective (although a trivial Kan fibration is because $\partial \Delta^0 = \emptyset \to \Delta^0$ is a generating cofibration. This corresponds nicely to the situation on the topological spaces level, where a Serre fibration is not necessarily surjective, but a trivial Serre fibration is.

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Sure, I'll buy that: $\emptyset \rightarrow \Delta^0$ should be a generating cofibration, but not an acyclic one. –  Aaron Mazel-Gee Nov 12 '12 at 16:53

Hirschhorn [Model categories and their localizations] writes:

Definition 7.10.8. If $f : X \to Y$ is a map of simplicial sets, then

  • $f$ is a weak equivalence if its geometric realization $\left| f \right| : \left| X \right| \to \left| Y \right|$ is a weak equivalence of topological spaces,
  • $f$ is a fibration if it is a Kan fibration, i.e. if it has the right lifting property with respect to the map $\Lambda [n, k] \to \Delta [n]$ for all $n > 0$ and $0 \le k \le n$, and
  • $f$ is a cofibration if it has the left lifting property with respect to all maps that are both fibrations and weak equivalences.

In other words, $\emptyset \hookrightarrow \Delta^0$ is not a horn inclusion... at least according to this definition.

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