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8.4.9 The Thompson Groupoid

In ยง8.4.7, we constructed an example of a homotopy idempotent endomorphism $e: X \rightarrow X$ which is not idempotent. Our construction (following Heller and Freyd) involved the Thompson group $\operatorname{Aut}_{ \mathrm{Dy}}( [0,1] )$ Our goal in this section is to show that this is no coincidence: there is a universal example of an $\infty $-category $\operatorname{\mathcal{C}}$ containing a homotopy idempotent endomorphism, whose structure can be described explicitly in terms of $\operatorname{Aut}_{ \mathrm{Dy}}( [0,1] )$. We begin with a variant of Definition 8.4.7.12.

Definition 8.4.9.1 (The Thompson Groupoid). We define a category $\mathrm{Dy}$ as follows:

  • The objects of $\mathrm{Dy}$ are closed intervals of the form $[0,s]$, where $s \geq 0$ is a dyadic rational number.

  • If $s,t \geq 0$ are dyadic rational numbers, then a morphism from $[0,s]$ to $[0,t]$ in the category $\mathrm{Dy}$ is a dyadic homeomorphism $[0,s] \xrightarrow {\sim } [0,t]$ (see Definition 8.4.7.9).

  • The composition law on $\mathrm{Dy}$ is given by composition of dyadic homeomorphisms (which is well-defined by virtue of Exercise 8.4.7.11.

It follows from Exercise 8.4.7.10 that the category $\mathrm{Dy}$ is a groupoid. We will refer to $\mathrm{Dy}$ as the Thompson groupoid.

Remark 8.4.9.2. The Thompson groupoid $\mathrm{Dy}$ contains exactly two isomorphism classes:

  • The isomorphism class of the degenerate interval $[0,0] = \{ 0 \} $, whose automorphism group is the trivial group.

  • The isomorphism class of the unit interval $[0,1]$, whose automorphism group is the Thompson group $\operatorname{Aut}_{ \mathrm{Dy} }( [0,1] )$ of Definition 8.4.7.12.

By virtue of Remark 8.4.9.2, the Thompson groupoid $\mathrm{Dy}$ is equivalent to the full subcategory spanned by the objects $\{ 0\} $ and $[0,1]$, which can be described explicitly in terms of the Thompson group $\operatorname{Aut}_{ \mathrm{Dy}}( [0,1] )$. However, allowing a larger class of intervals in the definition reveals some additional structure.

Construction 8.4.9.3 (Concatenation). Let $\mathrm{Dy}$ denote the Thompson groupoid. We define a functor $\circledast : \mathrm{Dy} \times \mathrm{Dy} \rightarrow \mathrm{Dy}$ as follows:

  • On objects, the functor $\circledast $ is given by the formula

    \[ [0,s] \circledast [0,t] = [0, s+t ]. \]
  • On morphisms, the functor $\circledast $ is given by the formula

    \[ (f \circledast g)(x) = \begin{cases} g(x) & \text{ if } 0 \leq x \leq s \\ f(x-s) + t & \text{ if } s \leq x \leq s+t. \end{cases} \]

We will refer to $\circledast : \mathrm{Dy} \times \mathrm{Dy} \rightarrow \mathrm{Dy}$ as the concatenation functor on the Thompson groupoid $\mathrm{Dy}$. Note that the operation $\circledast $ is strictly associative, and admits a (strict) unit given by the degenerate interval $\{ 0\} = [0,0]$. Consequently, $\circledast $ determines a strict monoidal structure on the category $\mathrm{Dy}$ (in the sense of Definition 2.1.2.1).

Notation 8.4.9.4. Let $B\mathrm{Dy}$ denote the (strict) $2$-category obtained by delooping $\mathrm{Dy}$ (see Example 2.2.0.8). Since $\mathrm{Dy}$ is a groupoid, $B\mathrm{Dy}$ is a $(2,1)$-category. It follows that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ is an $\infty $-category (Theorem 2.3.2.1). We can describe the low-dimensional simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ explicitly as follows:

  • The $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ has a unique object, which we will denote by $X$.

  • Morphisms from $\overline{X}$ to itself in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ can be identified with nonnegative dyadic rational numbers $s$ (corresponding to the closed interval $[0,s]$, regarded as an object of the Thompson groupoid $\mathrm{Dy}$).

  • Suppose we are given dyadic rational numbers $s,t,u \geq 0$. Then $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ with boundary indicated in the diagram

    \[ \xymatrix@R =50pt@C=50pt{ & \overline{X} \ar [dr]^{ s } & \\ \overline{X} \ar [ur]^{t} \ar [rr]^{u} & & \overline{X} } \]

    can be identified with dyadic homeomorphisms $[0, s+t ] \xrightarrow {\sim } [0,u]$.

Let $e: X \rightarrow X$ denote the morphism in $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ corresponding to the object $[0,1] \in \mathrm{Dy}$, and let $\sigma $ be the $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ corresponding to the dyadic homeomorphism

\[ [0,1] \circledast [0,1] = [0,2] \xrightarrow {\sim } [0,1] \quad \quad x \mapsto x/2. \]

Then the triple $(\overline{X}, \overline{e}, \overline{\sigma })$ can then be viewed as a partial idempotent $\iota : \operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ (see Example 8.4.8.3).

We can now formulate our main result:

Theorem 8.4.9.5. The partial idempotent $\iota : \operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ of Notation 8.4.9.4 is a categorical equivalence of simplicial sets.

Corollary 8.4.9.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the partial idempotent $\iota $ of Notation 8.4.9.4 induces a trivial Kan fibration of $\infty $-categories

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\leq 2}(\operatorname{Idem}), \operatorname{\mathcal{C}}). \]

Proof. Combine Theorem 8.4.9.5 with Corollary 4.5.5.19 (noting that $\iota $ is a monomorphism of simplicial sets). $\square$

Corollary 8.4.9.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an endomorphism $e: X \rightarrow X$. Then $e$ is homotopy idempotent if and only if there is a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} ) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \overline{e} ) = e$.

Example 8.4.9.8. Let $\mathrm{Dy}_{> 0}$ denote the full subcategory of the Thompson groupoid $\mathrm{Dy}$ spanned by the intervals $[0,s]$ where $s > 0$. Note that the action of $\mathrm{Dy}$ on itself (via concatenation) restricts to an action of $\mathrm{Dy}$ on the groupoid $\mathrm{Dy}_{> 0}$, and therefore determines a (strict) functor of $2$-categories

\[ B\mathrm{Dy} \rightarrow \{ \textnormal{Groupoids} \} \quad \quad X \mapsto \mathrm{Dy}_{> 0}. \]

Passing to nerves, we obtain a functor of $\infty $-categories

\[ \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} ) \rightarrow \operatorname{\mathcal{S}}\quad \quad X \mapsto \operatorname{N}_{\bullet }( \mathrm{Dy}_{> 0} ), \]

which carries the $1$-morphism $\overline{e}$ of $B \mathrm{Dy}$ to the homotopy idempotent endomorphism

\[ \operatorname{N}_{\bullet }( \mathrm{Dy}_{> 0} ) \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy}_{> 0} ) \quad \quad [0,s] \mapsto [0,1] \circledast [0,s] = [0, 1+s]. \]

Note that, up to isomorphism, this coincides with the homotopy idempotent endomorphism constructed in Proposition 8.4.7.14; this follows from the observation that the diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ B \operatorname{Aut}_{ \mathrm{Dy} }( [0,1] ) \ar [r]^-{\alpha } \ar [d]^{\sim } & B \operatorname{Aut}_{ \mathrm{Dy} }( [0,1] ) \ar [d]^{\sim } \\ \mathrm{Dy}_{> 0} \ar [r]^-{ [0,1] \circledast } & \mathrm{Dy}_{ > 0} } \]

commutes up to isomorphism (where the vertical maps are the inclusion functors and $\alpha $ is the homomorphism of Construction 8.4.7.13).

Our first goal is to reduce Theorem 8.4.9.5 to a more concrete statement about simplicial monoids.

Notation 8.4.9.9. Let $\operatorname{N}_{\leq 2}(\operatorname{Idem})$ denote the simplicial set described in Example 8.4.8.3. By virtue of Proposition 2.4.4.3, we can choose a simplicial category $\operatorname{\mathcal{C}}$ and a morphism of simplicial sets $u: \operatorname{N}_{\leq 2}(\operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}$ as a simplicial path category for $\operatorname{N}_{\leq 2}(\operatorname{Idem})$. The morphism $u$ carries the unique vertex $\widetilde{X}$ of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$ to an object $X \in \operatorname{\mathcal{C}}$. Set $E_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet }$, which we regard as a simplicial monoid. Evaluating the morphism $u$ on the nondegenerate edge $\widetilde{e}$ of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$, we obtain a morphism $e: X \rightarrow X$ in the category $\operatorname{\mathcal{C}}$, which we can view as a vertex of the simplicial set $E_{\bullet }$. Evaluating the morphism $u$ on the nondegenerate $2$-simplex of $\operatorname{N}_{\leq 2}( \operatorname{Idem})$, we obtain an edge $h: e^2 \rightarrow e$ in the simplicial monoid $E_{\bullet }$.

Let $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ denote the nerve of the Thompson groupoid. The concatenation functor of Notation 8.4.9.4 endows $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ with the structure of a simplicial monoid. We let $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$ denote the simplicial category given by delooping $\operatorname{N}_{\bullet }( \mathrm{Dy} )$ (Example 2.4.2.3), so that the homotopy coherent nerve of $B \operatorname{N}_{\bullet }( \mathrm{Dy} )$ can be identified with the Duskin nerve of the strict $2$-category $B \mathrm{Dy}$ (see Example 2.4.3.11). It follows that the partial idempotent $\iota $ of Notation 8.4.9.4 can be identified with a morphism from $\operatorname{N}_{\leq 2}( \operatorname{Idem})$ to $\operatorname{N}_{\bullet }^{\operatorname{hc}}( B \operatorname{N}_{\bullet }( \mathrm{Dy} ) )$, which factors unique as a composition

\[ \operatorname{N}_{\leq 2}( \operatorname{Idem}) \xrightarrow { u } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{N}_{\bullet }^{\operatorname{hc}}(F) } \operatorname{N}_{\bullet }^{\operatorname{hc}}( B \operatorname{N}_{\bullet }( \mathrm{Dy} ) ) \]

for some simplicial functor $F: \operatorname{\mathcal{C}}\rightarrow B \operatorname{N}_{\bullet }( \mathrm{Dy} )$, which we can identify with a homomorphism of simplicial monoids

\[ \varphi : E_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \xrightarrow {F} \operatorname{Hom}_{ B \operatorname{N}_{\bullet }( \mathrm{Dy} )}( F(X), F(X))_{\bullet } = \operatorname{N}_{\bullet }( \mathrm{Dy} ). \]

By construction, the homomorphism $\varphi $ carries the vertex $e$ to the object $[0,1] \in \mathrm{Dy}$, and the edge $h: e^2 \rightarrow e$ to the dyadic homeomorphism

\[ [0,1] \circledast [0,1] = [0,2] \xrightarrow {\sim } [0,1] \quad \quad x \mapsto x/2. \]

Remark 8.4.9.10. Using the universal property of the path category $\operatorname{\mathcal{C}}= \operatorname{Path}[ \operatorname{N}_{\leq 2}(\operatorname{Idem}) ]_{\bullet }$, it is not difficult to see that $E_{\bullet }$ is freely generated (as a simplicial monoid) by the vertex $e$ and the edge $h: e^2 \rightarrow e$. In particular, the homomorphism of simplicial monoids $\varphi : E_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy} )$ is uniquely determined by the requirement that it carries $e$ to the unit interval $[0,1] \in \mathrm{Dy}$ and $h$ to the dyadic homeomorphism $x \mapsto x/2$.

We will deduce Theorem 8.4.9.5 from the following:

Proposition 8.4.9.11. The homomorphism $\varphi : E_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy} )$ is a weak homotopy equivalence of simplicial sets.

Proof of Theorem 8.4.9.5 from Proposition 8.4.9.11. We wish to show that the partial idempotent

\[ \iota : \operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( B \operatorname{N}_{\bullet }(\mathrm{Dy} ) ) \]

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 1.2.4.2). 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 4.6.7.7. 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

\[ \varphi : E_{\bullet } = \operatorname{Hom}_{ \operatorname{Path}[\operatorname{N}_{\leq 2}(\operatorname{Idem})]}( X, X)_{\bullet } \rightarrow \operatorname{Hom}_{ B \operatorname{N}_{\bullet }( \mathrm{Dy} )}( F(X), F(X) ) = \operatorname{N}_{\bullet }( \mathrm{Dy} ), \]

which follows from Proposition 8.4.9.11. $\square$

We will deduce Proposition 8.4.9.11 from a more refined result, which characterizes the simplicial monoid $E_{\bullet }$ up to categorical equivalence (rather than merely up to weak homotopy equivalence).

Notation 8.4.9.12. Let $m$ and $n$ be nonnegative integers. We will say that a homeomorphism $f: [0,m] \xrightarrow {\sim } [0,n]$ is a dyadic contraction if, for every integer $0 \leq k < m$, the restriction of $f$ to the closed interval $[k, k+1]$ is given by the formula $f(x) = (x+a) / 2^ b$ for some integers $a$ and $b$ with $b \geq 0$.

We let $\mathrm{Dy}_{+}$ denote the subcategory of the Thompson groupoid $\mathrm{Dy}$ whose objects are intervals of the form $[0,m]$, where $m$ is a nonnegative integer, and whose morphisms are dyadic contractions.

Exercise 8.4.9.13. Show that the subcategory $\mathrm{Dy}_{+} \subset \mathrm{Dy}$ is well-defined: that is, the collection of dyadic contractions is closed under composition.

Warning 8.4.9.14. The category $\mathrm{Dy}_{+}$ of Notation 8.4.9.12 is not a groupoid. In fact, every isomorphism in the category $\mathrm{Dy}_{+}$ is an identity morphism.

Proposition 8.4.9.15. The homomorphism $\varphi : E_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy} )$ factors as a composition

\[ E_{\bullet } \xrightarrow { \varphi _{+} } \operatorname{N}_{\bullet }( \mathrm{Dy}_{+} ) \subset \operatorname{N}_{\bullet }( \mathrm{Dy} ), \]

where $\varphi _{+}$ is inner anodyne.

Proof of Proposition 8.4.9.11 from Proposition 8.4.9.15. By virtue of Proposition 8.4.9.15, 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

\[ [0,s] \xrightarrow {\sim } [0, 2^ k s] \quad \quad x \mapsto 2^{k} x \]

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. $\square$

The proof of Proposition 8.4.9.15 will require some preliminaries.

Notation 8.4.9.16. Let $n \geq 0$ be an integer and let $J$ be a subset of $\{ 1, 2, \cdots , n \} $. We let $b_{J}: [0,n] \xrightarrow {\sim } [0, n + |J| ]$ be the dyadic homeomorphism which is characterized by the following requirement: for every integer $1 \leq j \leq n$, the function $b_{I}$ is differentiable at every point $x \in (j-1, j)$, with derivative given by the formula

\[ b'_{J}(x) = \begin{cases} 1 & \text{if $j \notin J$} \\ 2 & \text{if $j \in J$.} \end{cases} \]

Note that, if this condition is satisfied, then the inverse homeomorphism $b_{J}^{-1}: [ 0, n + |J| ] \xrightarrow {\sim } [0,n]$ is a dyadic contraction. We say that a dyadic contraction is elementary if it has the form $b_{J}^{-1}$, for some integer $n \geq 0$ and some subset $J \subseteq \{ 1, 2, \cdots , n \} $.

Remark 8.4.9.17. Let $f: [0,m] \xrightarrow {\sim } [0,n]$ be a dyadic contraction. The following conditions are equivalent:

  • The dyadic contraction $f$ is elementary, in the sense of Notation 8.4.9.16.

  • For every point $x \in [0,m]$ where $f$ is differentiable, the derivative $f'(x)$ is either $1$ or $1/2$.

  • The dyadic contraction $f$ can be written as a concatenation $f_{1} \circledast f_2 \circledast \cdots \circledast f_ n$, where each $f_{i}$ is either the identity function $\operatorname{id}: [0,1] \xrightarrow {\sim } [0,1]$ or the homeomorphism

    \[ H: [0,2] \xrightarrow {\sim } [0,1] \quad \quad x \mapsto x/2. \]

Lemma 8.4.9.18. The collection of elementary dyadic contractions is a class of short morphisms for the $\infty $-category $\operatorname{N}_{\bullet }( \mathrm{Dy}_{+} )$, in the sense of Definition 6.2.5.4.

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 8.4.9.17). 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 8.4.9.16. 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 $(3)$.

$\square$

Proof of Proposition 8.4.9.15. 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 $\sigma :$

\[ [0, n_0] \xrightarrow {\sim } [0, n_1] \xrightarrow {\sim } \cdots \xrightarrow { \sim } [0, n_ m ] \]

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 8.4.9.18 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 8.4.9.9 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

\[ [0, n_0] \xrightarrow {\sim } [0, n_1] \xrightarrow {\sim } \cdots \xrightarrow { \sim } [0, n_ m ] \]

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

    \[ [0,1] \xrightarrow {\operatorname{id}} [0,1] \xrightarrow {\operatorname{id}} [0,1] \xrightarrow {\operatorname{id}} \cdots \xrightarrow {\operatorname{id}} [0,1]. \]
  • 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 1.2.6.3). 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 8.4.9.10.

  • 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 8.4.9.10.

$\square$