# On the natural (bigraded) homotopy groups of a simplicial object in a model category

$\def\mc{\mathcal} \def\sm{\wedge}$ This question stems from the Goerss-Hopkins paper Moduli Problems for Structured Ring Spectra. Let me begin by attempting to summarize the relevant framework -- this comes from the beginning of Chapter 3, on page 92.

Let $\mc{C}$ be a model category, and let $\mc{P}$ be a set of projectives defining the $\mc{P}$-resolution model stucture on $s\mc{C}$. (See Question 1 below.) Given $X \in s\mc{C}$ and for any $P\in \mc{P}$, define the $(n,P)$th natural homotopy group of $X$ by $\pi_{n,P}(X)=\pi_n(\mbox{map}(P,X))$ (using the derived mapping space). This is corepresentable in $\mbox{Ho}(s\mc{C})$ by $P \sm > \Delta^n/\partial\Delta^n$, which is defined as the pushout of the corner $\emptyset \otimes \ast \leftarrow P > \otimes \ast \rightarrow P \otimes > \Delta^n / \partial \Delta^n$. This smash product construction actually fits into an adjunction $$(-) \sm K : > \mc{C} / \emptyset \leftrightarrows s\mc{C} > : C_K,$$ where we define the right adjoint as follows. First, there is an adjunction $$(-)\otimes K:\mc{C} \leftrightarrows s\mc{C}:\mbox{hom}(K,-)_0 \stackrel{def}{=} > M_K(-)$$ (where the right adjoint is the $0$th object in the "hom object" of $s\mc{C}$ coming its simplicially contensored structure), and then we define $C_K$ via the pullback diagram $$\begin{array}{ccc} C_KX & > \rightarrow & M_KX \\ \downarrow & & > \downarrow \\ \emptyset & \rightarrow > & M_*X = X_0. \end{array}$$ In particular, write $C_nX=C_{\Delta^n / > \Lambda^n_0}X$ and $Z_nX=C_{\Delta^n / > \partial \Delta^n}X$. The "inclusion of the boundary" $\Delta^{n} > /\partial \Delta^{n} \rightarrow > \Delta^{n+1}/\Lambda^{n+1}_0$ induces the second map in a fibration sequence $$Z_{n+1} X \rightarrow C_{n+1}X \rightarrow > Z_{n}X$$ in the category $\mc{C}/\emptyset$ whenever $X \in s\mc{C}$ is Reedy fibrant. The neat result is that in this case, this in fact gives a "presentation" of the natural homotopy group $\pi_{n,P}(X)$ via the exact sequence $$[P,C_{n+1}X] \rightarrow > [P,Z_nX] \rightarrow \pi_{n,P}(X) > \rightarrow 0.$$

I have a few questions about the details and heuristics here.

Question 1: I'm confused about the extra conditions on $\mc{C}$ that might be necessary. They claim that the objects $P\in \mc{P}$ will be h-cogroup objects; this is what in particular gives a canonical map $P \rightarrow \emptyset$, which endows $\mbox{map}(P,X)$ with a basepoint. On the one hand, the definition of the $\mc{P}$-resolution model structure (see pages 24-26) starts out with the axiom that $\mc{P}$ be closed under suspension and desuspension, which suggests strongly that we're in a stable model category. But then it's silly to point out that the objects of $\mc{P}$ corepresent homotopy functors valued in abelian groups, since this is true of all objects. Even more, a stable model category is automatically pointed, which would mean that $\mc{C}/\emptyset$ is a totally vacuous thing to discuss; I'd consider this to be strong evidence that somehow $\mc{C}$ isn't meant to necessarily be stable. Of course the final application will be for $\mc{C}$ a category of spectra so this doesn't matter, but I've been pondering this long enough that the question has become of independent interest.

Question 2: Slightly further down (3.1.3 on page 95), they claim that the case $n=0$ of the result I cited yields an isomorphism $\pi_0([P,X]) \cong \pi_{0,P}(X)$ (where $[P,X]$ is a simplicial abelian group -- "simplicial" via $[P,X]_\bullet = [P,X_\bullet]$, and "abelian group" since supposedly $P$ is an (abelian, I guess) h-cogroup). However, unless I'm making a totally stupid mistake, I'm pretty sure that $Z_0X=C_{\Delta^0/\partial \Delta^0}X = C_*X = \emptyset$. I don't see how to make sense of this.

Question 3 (assuming $\mc{C}$ doesn't need to be stable): How should I understand taking a pullback over the initial object? If $\emptyset=*$, then I could consider $M_KX$ as "maps from $K$ to $X$" and $C_KX$ as "based maps from $K$ to $X$". This is really appealing; then I could view the "presentation" as saying that $\pi_{n,P}(X)$ is just $[P,-]$ applied to "based maps $S^n \rightarrow X$ mod based maps $D^{n+1} \rightarrow X$"! (It also makes sense that "based maps" should be adjoint to "smash product".) But as things stand, this isn't really honest. In fact, I don't think I've ever seen anyone take a pullback over an initial object (besides in the pointed case, of course). I can't seem to wrap my head around what $\mbox{lim}(\emptyset \rightarrow A \leftarrow B)$ should mean, besides "the last object $C$ over $B$ such that $C \rightarrow B \rightarrow A$ is trivial" (i.e. factors through $\emptyset$, obviously). This sounds an awful lot like the fiber of a map, but I'm not convinced that this analogy makes any sense. I'd be grateful to hear any geometric intuition that anyone can give here.

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Hi Aaron, I don't have time to write any long comments right now -- but if you are serious about looking for unstable results you should definitely take a look at the paper of Bousfield's that their resolution model structures are based on (arxiv.org/abs/math.AT/0312531) and dualize everything. –  Tyler Lawson Oct 22 '12 at 1:08
(Not to mention the Dwyer-Kan-Stover paper, of course.) –  Tyler Lawson Oct 22 '12 at 1:09
Thanks for the suggestion. I've already checked out Dwyer-Kan-Stover briefly, but it didn't seem to resolve these questions. And no, I don't care strongly about the unstable version -- my hands are already pretty full with trying to understand this paper! -- I've just become curious about it in the course of things. –  Aaron Mazel-Gee Oct 22 '12 at 1:28

Dear Aaron,

For question 1, you're correct in that all objects are cogroup objects. This isn't necessary for the original version from Bousfield's paper--only one of the suspensions is--and there you do need to make sure that your objects you're using to define the model structure really are cogroup objects. (You can make use of this in the stable setting too. Paul Goerss once pointed out to me that you can restrict your cogroup objects to high-dimensional spheres to produce Postnikov towers out of the resolution structure).

For question 2, my guess is that you're running into issues because $\Delta^0 / \partial \Delta^0$, the space obtained by taking a single point and collapsing the empty set to a single point, isn't $\ast$; by convention in homotopy theory, $X / \emptyset = X_+$.

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I see; thanks for the clarification. And yes, I suppose that convention $X/\emptyset = X_+$ makes sense: the rule is to take the subspace and identify it all with a (new) point. I'll leave this open for a few days in case anyone has anything to say about my third question. –  Aaron Mazel-Gee Oct 24 '12 at 0:40
@Aaron: So far as your last question, defining pullbacks over the initial object is working in a category of objects-over-the-initial-object, and so it's really some kind of pointed context anyway - I couldn't figure out anything profound to say. –  Tyler Lawson Oct 27 '12 at 12:34