# Difficulty with lemma $7.4$ of Goerss-Jardine: a simplex $\alpha$ is nullhomotopic to $v$ iff. there is a simplex with boundary $(v,\cdots,v,\alpha)$

We are given a nonempty fibrant simplicial set $$X$$ (a Kan complex) and $$v\in X_0$$ is any vertex. We are $$\alpha\in X_n$$ and the map $$\alpha:\Delta^n\to X$$ is homotopic to $$v:\Delta^n\overset{!}{\longrightarrow}\Delta^0\overset{v}{\longrightarrow}X$$ relative to $$\partial\Delta^n$$ (throughout, I will abuse notation as Goerss and Jardine do, by letting $$v$$ denote any degenerate image of $$v$$ in any $$X_n$$).

Let $$h:v\simeq\alpha:\Delta^n\times\Delta^1\to X$$ be such a homotopy. We have $$h\big|_{\partial\Delta^n\times\Delta^1}=v\circ\pi:\partial\Delta^n\times\Delta^1\to\partial\Delta^n\to X$$ and $$d_i\alpha=v$$ for all $$0\le i\le n$$ by definition of relative homotopy.

The task is to show there exists $$\omega\in X_{n+1}$$ with boundary $$\partial\omega=(d_0\omega,d_1\omega,\cdots,d_n\omega,d_{n+1}\omega)=(v,v,\cdots,v,\alpha)$$.

I'm having surprising difficulty with this. I know that we can exploit anodyne extensions to 'fill horns' or similar, e.g. I considered in particular the following: $$(\Delta^1\times\Lambda^n_i)\cup(\partial\Delta^1\times\Delta^n)\hookrightarrow\Delta^1\times\Delta^n\\(\Delta^1\times\partial\Delta^n)\cup(\partial\Delta^1\times\Delta^n)\hookrightarrow\Delta^1\times\Delta^n\\(\Delta^1\times\partial\Delta^n)\cup(\Lambda^1_0\times\Delta^n)\hookrightarrow\Delta^1\times\Delta^n$$To generate new maps $$\Delta^1\times\Delta^n\to X$$. This was to no avail, because the only sensible maps to put on the components would have to use $$h$$ somewhere, but $$h$$ acts trivially or uninterestingly on each such one (e.g. $$h$$'s restriction to $$\partial\Delta^1\times\Delta^n$$ is the map $$(v,\alpha)$$ and this map exists regardless of whether or not $$h$$ does). In short, I did not see a way to exploit $$h$$ here.

I might consider similar extensions but with $$n+1$$ instead of $$n$$. However, a resulting map $$\Delta^1\times\Delta^{n+1}\to X$$ does not induce an interesting $$\omega:\Delta^{n+1}\to X$$, unless I'm mistaken.

Another approach might be to directly use $$h$$ in some composite $$\omega:\Delta^{n+1}\overset{p}{\longrightarrow}\Delta^1\times\Delta^n\overset{h}{\longrightarrow}X$$. In order for this to have $$d_{n+1}\omega=\alpha$$, we need $$p(\delta_{n+1})$$ equal to $$(1,\mathrm{id}_n)$$, and for $$d_i\omega=v$$ for all $$i\le n$$ we need to guarantee $$p(\delta_i)$$ is either in $$\{0\}\times\Delta^n$$ or in $$\Delta^1\times\partial\Delta^n$$. To make the two compatible, it can only be that $$p(\delta_i)$$ is in $$\Delta^1\times\partial\Delta^n$$.

But, examining all possible $$(n+1)$$-simplices of $$\Delta^1\times\Delta^n$$, I am fairly sure no such $$p$$ exists. I don't know what other tools I have at my disposal, so I'd really appreciate any hints or answers.

EDIT: Here is an explicit version of the below answer.

let $$h:v\simeq\alpha$$ rel. $$\partial\Delta^n$$. The maps $$v\pi:\Delta^1\times\Delta^n\to\Delta^n\to\Delta^0\to X$$ and $$h:\Delta^1\times\Delta^n$$ determine, by a known coequaliser, a unique map $$(v\pi,v\pi,\cdots,h):\Delta^1\times\partial\Delta^{n+1}\to X$$ since $$h$$ restricts to $$v$$ on its boundaries (relative homotopy condition). $$v\pi:\{0\}\times\Delta^{n+1}\to\Delta^0\to X$$ is compatible with $$(v,\cdots,h)$$ on the common intersection $$\{0\}\times\partial\Delta^n$$ since $$h$$ restricts to $$v$$ on $$\{0\}\times(-)$$, so we have a well defined map: $$(\Delta^1\times\partial\Delta^{n+1})\cup(\{0\}\times\Delta^{n+1})\overset{((v\pi,\cdots,v\pi,h),v\pi)}{\longrightarrow}X$$As $$X$$ is fibrant and the (obvious) inclusion $$(\Delta^1\times\partial\Delta^{n+1})\cup(\{0\}\times\Delta^{n+1})\hookrightarrow\Delta^1\times\Delta^{n+1}$$ is anodyne, there is an extension of this map to $$\psi:\Delta^1\times\Delta^{n+1}\to X$$.

If $$\iota:\Delta^{n+1}\hookrightarrow\Delta^1\times\Delta^{n+1}$$ maps $$\sigma\mapsto(1,\sigma)$$, then let $$\omega:=\psi\circ\iota:\Delta^{n+1}\to X$$ (considered also as an $$(n+1)$$-simplex). If $$i\le n$$, then $$d_i\omega=\omega(\delta_i)=v\pi((1,1))=v$$ and $$d_n\omega=\omega(\delta_n)=h(1,1)=\alpha(1)=\alpha$$ as required.

The first approach should work.

Map one side of the prism $$\Delta^1 \times \Delta^{n+1}$$ via $$h$$ and all others via the constant map equal to $$v$$. Map the button to $$v$$ as well. This produces a well-defined map from a "cup" to $$X$$. Extend it to a map $$\Delta^1 \times \Delta^{n+1} \to X.$$ Restrict to the top of the prism for the desired simplex.

• I see I was too quick to dismiss the idea of filling in a map $\Delta^1\times\Delta^{n+1}\to X$ (you should replace $n$ with $(n+1)$ for the benefit of others). Nov 23, 2022 at 9:49
• @FShrike Oh, thanks Nov 23, 2022 at 19:32

This can be usefully generalised, and I think this generalisation is what Goerss-Jardine wanted us to use (for the later proof, also left as an exercise).

Lemma:

Let $$X$$ be any nonempty fibrant simplicial set, $$v\in X_0$$ any distinguished vertex and $$n\in\Bbb N$$, $$0\le i\le n+1$$ arbitrary integers. For $$\alpha\in X_n$$ an $$n$$-simplex with boundary $$(v,v,\cdots,v)$$, $$\alpha$$ is homotopic to $$v$$ rel. $$\partial\Delta^n$$ iff. there exists an $$\omega\in X_{n+1}$$ with boundary $$\underset{(0,1,\cdots,i-1,i,i+1,\cdots,n+1)}{\underbrace{(v,\cdots,v,\alpha,v,\cdots,v)}}$$.

Proof:

Let $$X,\alpha,i,n$$ be given and assume existence of such an $$\omega$$. There is a map: $$\large(\Delta^{n+1}\times\partial\Delta^1)\cup(\Lambda^{n+1}_i\times\Delta^1)\overset{((\omega,v),(v,\cdots,v,-,v,\cdots,v))}{\longrightarrow}X$$

Since $$d_j\omega=v$$ for $$j\neq i$$. Since $$X$$ is fibrant, there is a lift of this map along the anodyne extension $$(\Delta^{n+1}\times\partial\Delta^1)\cup(\Lambda^{n+1}_i\times\Delta^1)\hookrightarrow\Delta^{n+1}\times\Delta^1$$ given by some $$\psi:\Delta^{n+1}\times\Delta^1\to X$$.

In fact, $$\psi$$ is a homotopy $$\omega\simeq v$$, but this homotopy is not necessarily relative to $$\partial\Delta^{n+1}$$. Anyway, define $$h:\Delta^n\times\Delta^1\to X$$ as the composite $$\psi\circ((\delta_i\circ(-))\times1)$$.

We have that $$h$$ restricts to $$v$$ on $$\partial\Delta^n\times\Delta^1$$ since $$\delta_i\circ f\in\Lambda^{n+1}_i$$ for any non-surjective $$f$$ and $$h$$ restricts to $$(d_i\omega,v)$$ on $$\Delta^n\times\partial\Delta^1$$, so $$h:\alpha\simeq v$$ is a relative homotopy.

Conversely suppose such an $$h$$ exists, $$\alpha\simeq v$$ rel. $$\partial\Delta^n$$. There is a map: $$\large(\Delta^{n+1}\times\{1\})\cup(\partial\Delta^{n+1}\times\Delta^1)\overset{\overset{(0,\cdots,i-1,i,i+1,\cdots,n+1)}{\overbrace{(v,(v,\cdots,v,h,v,\cdots,v))}}}{\longrightarrow}X$$

Valid for reasons described in my question. There is a lift of this map along the anodyne extension $$(\Delta^{n+1}\times\{1\})\cup(\partial\Delta^{n+1}\times\Delta^1)\hookrightarrow\Delta^{n+1}\times\Delta^1$$ given by some $$\psi:\Delta^{n+1}\times\Delta^1\to X$$. Define $$\omega:=\psi\circ(-,0):\Delta^{n+1}\hookrightarrow\Delta^{n+1}\times\Delta^1\to X$$.

The boundary of this simplex $$\omega$$ is precisely $$(v,\cdots,v,\alpha,v,\cdots,v)$$ as required.