Kerodon

Proof. Let $\sigma = (\sigma _{-}, \sigma _{+})$ be a nondegenerate $n$-simplex of the product $S^{-} \times S^{+}$. Using Proposition 1.1.3.8, we see that $\sigma _{-}$ and $\sigma _{+}$ admit factorizations

where $\tau _{-}$ and $\tau _{+}$ are nondegenerate, so that $n_{-} \leq k_{-}$ and $n_{+} \leq k_{+}$. It follows that $\sigma $ factors as a composition

The nondegeneracy of $\sigma $ guarantees that the map of partially ordered sets $[n] \xrightarrow { (\alpha _{-}, \alpha _{+} )} [n_{-} ] \times [n_{+} ]$ is a monomorphism, so that $n \leq n_{-} + n_{+} \leq k_{-} + k_{+}$. $\square$

Proof. Let $m$ be the smallest nonnegative integer for which $\sigma $ can be factored as a composition $\Delta ^{n} \xrightarrow {\alpha } \Delta ^{m} \xrightarrow {\tau } S$. It follows from the minimality of $m$ that $\alpha $ must induce a surjection of linearly ordered sets $[n] \twoheadrightarrow [m]$ (otherwise, we could replace $[m]$ by the image of $\alpha $) and that the $m$-simplex $\tau $ is nondegenerate. This proves the existence of the desired factorization.

We now establish uniqueness. Suppose we are given another factorization of $\sigma $ as a composition $\Delta ^{n} \xrightarrow {\alpha '} \Delta ^{m'} \xrightarrow {\tau '} S$, and assume that $\alpha '$ induces a surjection $[n] \twoheadrightarrow [m']$. We first claim that, for any pair of integers $0 \leq i < j \leq n$ satisfying $\alpha '(i) = \alpha '(j)$, we also have $\alpha (i) = \alpha (j)$. Assume otherwise. Then $\alpha $ admits a section $\beta : \Delta ^{m} \hookrightarrow \Delta ^ n$ whose images include $i$ and $j$. We then have

Our assumption that $\alpha '(i) = \alpha '(j)$ guarantees that the map $(\alpha ' \circ \beta ): \Delta ^{m} \rightarrow \Delta ^{m'}$ is not injective on vertices, contradicting our assumption that $\tau $ is nondegenerate.

It follows from the preceding argument that $\alpha $ factors uniquely as a composition $\Delta ^{n} \xrightarrow {\alpha '} \Delta ^{m'} \xrightarrow {\alpha ''} \Delta ^{m}$, for some morphism $\alpha '': \Delta ^{m'} \rightarrow \Delta ^ m$ (which is also surjective on vertices). Let $\beta '$ be a section of $\alpha '$, and note that we have

Consequently, if the simplex $\tau '$ is nondegenerate, then $\alpha ''$ must also be injective on vertices. It follows that $m' = m$ and $\alpha ''$ is the identity map, so that $\alpha = \alpha '$ and $\tau = \tau '$. $\square$

Proof. The implication $(4) \Rightarrow (3)$ is trivial, the implication $(3) \Rightarrow (2)$ follows from Example 1.1.3.2, and the implication $(2) \Rightarrow (1)$ follows from Remarks 1.1.3.3 and 1.1.3.4. It will therefore suffice to show that $(1)$ implies $(4)$. Assume that $S$ has dimension $\leq k$, and let $T$ denote the colimit $\varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S, \leq k} } \Delta ^ n$; we wish to show that the tautological map $f: T \rightarrow S$ is an isomorphism of simplicial sets. Since $S$ has dimension $\leq k$, it follows immediately from the construction that the image of $f$ contains every nondegenerate simplex of $S$. Applying Remark 1.1.2.6, we deduce that $f$ is an epimorphism of simplicial sets. We will complete the proof by showing that $f$ is injective. Let $\tau $ and $\tau '$ be $\ell $-simplices of $T$ satisfying $f(\tau ) = f(\tau ')$; we wish to show that $\tau = \tau '$. Choose an object $( [n] , \sigma ) \in \operatorname{{\bf \Delta }}_{S, \leq k}$ and a lift of $\tau $ to an $\ell $-simplex $\widetilde{\tau }$ of $\Delta ^ n$, which we can identify with a nondecreasing function from $[\ell ]$ to $[n]$. Note that $\widetilde{\tau }$ factors uniquely as a composition $[\ell ] \xrightarrow { \alpha } [m] \xrightarrow { \beta } [n]$, where $\alpha $ is surjective and $\beta $ is injective. Replacing $n$ by $m$ and $\sigma $ by the associated $\ell $-simplex of $S$, we can reduce to the case where $\widetilde{\tau }: [\ell ] \twoheadrightarrow [n]$ is a surjection. Using Proposition 1.1.3.8, we can factor $\sigma $ as a composition

where $\gamma $ is surjective and $\rho $ is a nondegerate $p$-simplex of $S_{\bullet }$. Replacing $([n], \sigma )$ by $( [p], \rho )$ and $\widetilde{\tau }$ by the composition $\gamma \circ \widetilde{\tau }$, we can further assume that $\sigma $ is a nondegenerate $n$-simplex of $S_{\bullet }$. Similarly, we may assume that $\tau '$ lifts to an $m$-simplex $\widetilde{\tau }'$ of $\Delta ^{n'}$, for some object $([n'], \sigma ' )$ of $\operatorname{{\bf \Delta }}_{S, \leq k}$ where $\sigma '$ is nondegenerate and $\widetilde{\tau }': [m] \twoheadrightarrow [n']$ is surjective. We then have an equality

The uniqueness assertion of Proposition 1.1.3.8 then implies that $([n], \sigma ) = ( [n'], \sigma ' )$ and $\widetilde{\tau } = \widetilde{\tau }'$, so that $\tau $ and $\tau '$ are the same $m$-simplex of $T$. $\square$