Kerodon

Proof of Proposition 8.5.9.15. Let $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}}$ denote the simplicial subset of $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )$ whose $m$-simplices are diagrams of dyadic contractions $\sigma :$

for which the composite map $[0, n_0] \xrightarrow {\sim } [0,n_ m]$ is an elementary dyadic contraction (note that this guarantees that each intermediate composition $[0,n_ i] \xrightarrow {\sim } [0, n_ j]$ is also elementary). It follows from Lemma 8.5.9.18 and Theorem 6.2.5.10 that the inclusion map $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}} \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )$ is inner anodyne. We will complete the proof by showing that the morphism $\varphi : E_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy} )$ induces of Notation 8.5.9.9 induces an isomorphism of $E_{\bullet }$ with the simplicial subset $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}} \subseteq \operatorname{N}_{\bullet }( \mathrm{Dy} )$.

Fix an integer $m \geq 0$, so that $\varphi $ induces a monoid homomorphism $\varphi _{m}: E_{m} \rightarrow \operatorname{N}_{m}( \mathrm{Dy} )$. We wish to show that $\varphi _{m}$ is a monomorphism, whose image is the subset $\operatorname{N}_{m}( \mathrm{Dy}_{+} )^{\mathrm{short}} \subseteq \operatorname{N}_{m}( \mathrm{Dy} )$. Note that $\operatorname{N}_{m}( \mathrm{Dy}_{+} )^{\mathrm{short}}$ is closed under concatenation, and therefore inherits the structure of a monoid. Let us say that an $m$-simplex $\sigma $ of $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}}$ is indecomposable if it corresponds to a diagram of dyadic contractions

with $n_ m = 1$. In this case, we define the index of $\sigma $ to be the smallest integer $k$ such that $n_ k = 1$. For every integer $0 \leq k \leq m$, the simplicial set $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}}$ has a unique indecomposable $m$-simplex $\sigma _ k$ of index $k$, which can be described explicitly as follows:

• If $k = 0$, then $\sigma _ k$ is the diagram of identity morphisms

  \[ \empty [0,1] \xrightarrow {\operatorname{id}} [0,1] \xrightarrow {\operatorname{id}} [0,1] \xrightarrow {\operatorname{id}} \cdots \xrightarrow {\operatorname{id}} [0,1]. \]

• If $k > 0$, then the diagram $\sigma _ k$ has the form

  \[ [0,2] \xrightarrow {\operatorname{id}} \cdots \xrightarrow {\operatorname{id}} [0,2] \xrightarrow {x \mapsto x/2} [0,1] \xrightarrow {\operatorname{id}} \cdots \xrightarrow {\operatorname{id}} [0,1]. \]

Moreover, every $m$-simplex of $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}}$ can be written uniquely as a concatenation of indecomposable $m$-simplices of $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}}$: that is, $\operatorname{N}_{m}( \mathrm{Dy}_{+} )^{\mathrm{short}}$ can be identified with the free monoid generated by the set $\{ \sigma _0, \sigma _1, \cdots , \sigma _ m \} $.

Let $\operatorname{\mathcal{C}}_{\bullet } = \operatorname{Path}[ \operatorname{N}_{\leq 2}( \operatorname{Idem}) ]_{\bullet }$ denote the simplicial path category of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$. Theorem 2.4.4.10 supplies an identification of $\operatorname{\mathcal{C}}_{m}$ with the path category $\operatorname{Path}[ G ]$, where $G$ is a directed graph having a single vertex $X$ (corresponding to the unique vertex of the simplicial set $\operatorname{N}_{\leq 2}(\operatorname{Idem})$. It follows that $E_{m} = \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{m} }(X,X)$ can be identified with the free monoid generated by the set of edges $\operatorname{Edge}(G)$ (Example 1.3.7.3). It will therefore suffice to prove the following:

$(\ast _ m)$

The monoid homomorphism $\varphi _{m}$ induces a bijection from the set $\operatorname{Edge}(G)$ to the collection $\{ \sigma _0, \sigma _1, \cdots , \sigma _ m \} $ of indecomposable $m$-simplices of $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )^{\mathrm{short}}$.

To prove $(\ast _ m)$, we recall that $\operatorname{Edge}(G)$ can be identified with the set of pairs $( \tau , \overrightarrow {I} )$, where $\tau $ is a nondegenerate simplex of $\operatorname{N}_{\leq 2}(\operatorname{Idem})$ of dimension $n > 0$ and $\overrightarrow {I} = (I_0 \supseteq I_{1} \supseteq \cdots \supseteq I_{m-1} \supseteq I_ m )$ is a chain of subsets of $[n] = \{ 0 < 1 < \cdots < n \} $ satisfying $I_0 = [n]$ and $I_ m = \{ 0, n \} $. We consider two possibilities:

• The simplex $\tau $ has dimension $n=1$. In this case, both $\tau $ and $\overrightarrow {I}$ are uniquely determined. We claim that the homomorphism $\varphi _{m}$ carries $(\tau , \overrightarrow {I})$ to the indecomposable $m$-simplex $\sigma _0$. To prove this, we can invoke our assumption that $\varphi $ is a morphism of simplicial monoids to reduce to the case $m=0$, in which case it reduces to the identity $\varphi (e) = [0,1]$ of Remark 8.5.9.10.

• The simplex $\tau $ has dimension $n=2$. In this case, $\tau $ is again uniquely determined, and the chain $\overrightarrow {I}$ is determined by a single integer $1 \leq k \leq m$, given by the formula $k = \mathrm{min} \{ j: I_{j} = \{ 0,2\} \} $. We claim that the homomorphism $\varphi _{m}$ carries $( \tau , \overrightarrow {I} )$ to the indecomposable $m$-simplex $\sigma _ k$. To prove this, we can again invoke our assumption that $\varphi $ is a morphism of simplicial monoids to reduce to the case $m=1$, in which case it reduces to the assertion that $\varphi $ carries the edge $h: e^2 \rightarrow e$ of $E_{\bullet }$ to the dyadic contraction

  \[ [0,2] \xrightarrow {\sim } [0,1] \quad \quad x \mapsto x/2; \]

  see Remark 8.5.9.10.