The category of simplices $\Delta$ has a terminal object $[0]$, hence its nerve is contractible. What can be said about the nerve of its subcategory $\Delta_{\mathrm{mono}}$ which contains only the coface maps?
$\begingroup$
$\endgroup$
asked Dec 12, 2011 at 18:02
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
Let $C: \Delta_{\mathrm{mono}}\to \Delta_{\mathrm{mono}}$ be the "cone" functor, given on objects by $C([p])=[p+1]$, and on morphisms by $C(\delta)(0)=0$ and $C(\delta)(i) = \delta(i-1)+1$. Then there are natural monomorphisms $$[p] \to C([p]) \leftarrow [0]$$ which give a zig-zag of natural transformations relating the identity functor on $\Delta_{\mathrm{mono}}$ to a constant functor. So its got contractible nerve.
answered Dec 12, 2011 at 18:33
-
1$\begingroup$ Excellent, thank you! I figured there should be some simple argument like that. $\endgroup$ Commented Dec 12, 2011 at 19:42
-
5$\begingroup$ If I am not mistaken, this is the same proof as the one given by Maltsiniotis in "La Théorie de l'homotopie de Grothendieck", page 68 of math.jussieu.fr/~maltsin/ps/prstnew.pdf. He goes on to prove that not only is this category contractible, but this is also a weak test category. You can also have a look at section 1.8 of the document. (This comment may be slightly off-topic, but I thought people interested in this question might be interested to know that.) $\endgroup$ Commented Dec 12, 2011 at 20:10
-
$\begingroup$ @JonathanChiche Maybe the version has changed (and this is mostly for my own benefit), but it now seems that this is Lemma 1.7.24 on page 72 (page 78 of the PDF). Maltsiniotis denotes the semisimplex category $\Delta'$ from Example 1.7.23, but doesn't seem to give it a name, which makes it a bit hard to search for. The proof that $\Delta'$ is a test category is the succeeding Prop 1.7.25. $\endgroup$ Commented May 25, 2021 at 21:12