Kerodon

Proof. Combine Theorem 8.5.9.5 with Corollary 4.5.5.19 (noting that $\iota $ is a monomorphism of simplicial sets). $\square$

Proof of Theorem 8.5.9.5 from Proposition 8.5.9.11. We wish to show that the partial idempotent

is a categorical equivalence of simplicial sets. Since the category $\mathrm{Dy}$ is a groupoid, the simplicial monoid $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ is a Kan complex (Proposition 1.3.5.2). It follows that the simplicial category $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$ is locally Kan. Invoking Theorem , we are reduced to showing that $\iota $ induces a weak equivalence of simplicial categories $F: \operatorname{Path}[ \operatorname{N}_{\leq 2}(\operatorname{Idem}) ]_{\bullet } \rightarrow B \operatorname{N}_{\bullet }( \mathrm{Dy} )$, in the sense of Definition 4.6.8.7. Write $X$ for the unique object of the path category $\operatorname{Path}[ \operatorname{N}_{\leq 2}(\operatorname{Idem}) ]_{\bullet }$, so that $F(X)$ is the unique object of $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$. We are then reduced to showing that $F$ induces a weak homotopy equivalence of simplicial monoids

which follows from Proposition 8.5.9.11. $\square$

Proof of Proposition 8.5.9.11 from Proposition 8.5.9.15. By virtue of Proposition 8.5.9.15, it will suffice to show that the inclusion of categories $\mathrm{Dy}_{+} \hookrightarrow \mathrm{Dy}$ induces a weak homotopy equivalence of simplicial sets $U: \operatorname{N}_{\bullet }( \mathrm{Dy}_{+} ) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Dy} )$. Using Quillen's Theorem A (Example 7.2.3.3), we are reduced to proving the following: for every object $[0,s] \in \mathrm{Dy}$, the category $\operatorname{\mathcal{A}}= \mathrm{Dy}_{+} \times _{ \mathrm{Dy} } \mathrm{Dy}_{ [0,s] / }$ has weakly contractible nerve. We can describe the category $\operatorname{\mathcal{A}}$ more concretely as follows:

• The objects of $\operatorname{\mathcal{A}}$ are dyadic homeomorphisms $f: [0,s] \xrightarrow {\sim } [0,m]$, where $m$ is an integer.

• Let $f: [0,s] \xrightarrow {\sim } [0,m]$ and $g: [0,s] \xrightarrow {\sim } [0,n]$ be dyadic homeomorphisms. Then there is a morphism from $f$ to $g$ (in the category $\operatorname{\mathcal{A}}$) if and only if the homeomorphism $(g \circ f^{-1}): [0,m] \rightarrow [0,n]$ is a dyadic contraction. If this condition is satisfied, then the morphism is unique.

It follows that the category $\operatorname{\mathcal{A}}$ can be viewed as a partially ordered set. Moreover, every finite subset of $\operatorname{\mathcal{A}}$ has a lower bound, given by the dyadic homeomorphism

for some integer $k \gg 0$. It follows that the category $\operatorname{\mathcal{A}}^{\operatorname{op}}$ is filtered (Exercise 7.2.4.2), so that $\operatorname{N}_{\bullet }( \operatorname{\mathcal{A}})$ is weakly contractible by virtue of Proposition 7.2.4.9. $\square$

Proof. Let $S$ denote the collection of all elementary dyadic contractions. We verify that $S$ satisfies conditions $(1)$ through $(4)$ of Definition 6.2.5.4:

$(1)$

For every integer $n \geq 0$, the identity morphism $\operatorname{id}: [0,n] \xrightarrow {\sim } [0,n]$ is an elementary dyadic contraction. This is immediately from the definitions.

$(2)$

Suppose we are given a commutative diagram of dyadic contractions

\[ \xymatrix@R =50pt@C=50pt{ & [0,m] \ar [dr]^{g} & \\[0,k] \ar [ur]^{ f } \ar [rr]^{h} & & [0,n]. } \]

Assume that $g$ and $h$ are elementary; we wish to show that $f$ is also elementary (in fact, the assumption that $g$ is elementary will not be needed). Choose a point $x \in [0,k]$ at which $f$ is differentiable; we wish to show that $f'(x) \geq 1/2$ (see Remark 8.5.9.17). Replacing $x$ by a nearby point if necessary, we may assume that $g$ is differentiable at the point $y = f(x)$. Since $g$ is a dyadic contraction and $h$ is an elementary dyadic contraction, we have $g'(y) \leq 1$ and $h'(x) \geq 1/2$. Applying the chain rule, we obtain inequalities $f'(x) \geq f'(x) \cdot g'(y) = h'(x) \geq 1/2$.

$(3)$

Let $f: [0,m] \xrightarrow {\sim } [0,n]$ be a dyadic contraction. We wish to show that $f$ admits an $S$-optimal factorization (in the sense of Definition 6.2.5.1). Let $P$ denote the collection of all subsets $\{ 1, 2, \cdots , n \} $ having the property that the composition $(b_ J \circ f): [0,m] \rightarrow [0, n+|J|]$ is a dyadic contraction; here $b_ J$ denotes the dyadic homeomorphism introduced in Notation 8.5.9.16. Unwinding the definitions, we can identify $P$ with the set of factorizations $f = s \circ g$, where $g$ is a dyadic contraction and $s$ is an elementary dyadic contraction (the identification carries a set $J \in P$ to the pair $(s,g) = ( b_{J}^{-1}, b_{J} \circ f)$). Under this identification, a factorization $f = s \circ g$ is $S$-optimal if and only if $J$ is a largest element of $P$. We conclude by observing that $P$ has a largest element $J_{\mathrm{max}}$, given by the collection of those integers $j \in \{ 1,2, \cdots , n \} $ having the property that the inverse homeomorphism $f^{-1}$ has derivative $\geq 2$ at every point $x \in [j-1,j]$ where $f^{-1}$ is differentiable (alternatively, $J_{\mathrm{max}}$ can be described as the set of integers $1 \leq j \leq n$ which satisfy $f^{-1}(j) > f^{-1}( j-1) + 1$).

$(4)$

Let $f: [0,m] \rightarrow {\sim } [0,n]$ be a dyadic contraction. Let us define the length of $f$ to be the smallest nonnegative integer $k$ such that $f'(x) \geq 1/2^ k$ for every point $x \in [0,m]$ where $f$ is differentiable. We claim that, if this condition is satisfied, then $f$ can be written as a composition $s_1 \circ s_2 \circ \cdots \circ s_ k$, where each $s_ i$ is an elementary dyadic contraction. Our proof proceeds by induction on $k$. If $k = 0$, then $f$ is an identity morphism and there is nothing to prove. Let us therefore assume that $k > 0$, and let $f = s_1 \circ g$ be an $S$-optimal factorization of $f$. We claim that $g$ can be written as a composition of elementary contractions $s_2 \circ \cdots \circ s_ k$. By virtue of our inductive hypothesis, it will suffice to show that $g$ has length $k-1$, which follows from the proof of $(3)$.

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.