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1.1.3 Dimensions of Simplicial Sets

We now introduce an important complexity measure for simplicial sets.

Definition 1.1.3.1. Let $S$ be a simplicial set and let $k$ be an integer. We will say that $S$ has dimension $\leq k$ if every $n$-simplex of $S$ is degenerate for $n > k$. If $k \geq 0$, we say that $S$ has dimension $k$ if it has dimension $\leq k$ but does not have dimension $\leq k-1$. We say that $S$ is finite-dimensional if it has dimension $\leq k$ for some $k \gg 0$.

Example 1.1.3.2. For each $n \geq 0$, the standard simplex $\Delta ^ n$ has dimension $n$.

Remark 1.1.3.3. Let $S$ be the coproduct of a collection of simplicial sets $\{ S(a) \} _{a \in A}$. Then $S$ has dimension $\leq k$ if and only if each $S(a)$ has dimension $\leq k$.

Remark 1.1.3.4. Let $f: S \twoheadrightarrow T$ be an epimorphism of simplicial sets. If $S$ has dimension $\leq n$, then $T$ has dimension $\leq n$.

Remark 1.1.3.5. Let $k$ be an integer. If a simplicial set $S$ has dimension $\leq k$, then every simplicial subset of $S$ has dimension $\leq k$ (see Remark 1.1.2.5).

Proposition 1.1.3.6. Let $S^{-}$ and $S^{+}$ be simplicial sets having dimensions $\leq k_{-}$ and $\leq k_{+}$, respectively. Then the product $S^{-} \times S^{+}$ has dimension $\leq k_{-} + k_{+}$.

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

$\Delta ^{n} \xrightarrow { \alpha _{-} } \Delta ^{n_{-}} \xrightarrow { \tau _{-} } S^{-} \quad \quad \Delta ^{n} \xrightarrow { \alpha _{+} } \Delta ^{n_{+}} \xrightarrow { \tau _{+} } S^{+},$

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

$\Delta ^{n} \xrightarrow { (\alpha _{-}, \alpha _{+} ) } \Delta ^{n_{-}} \times \Delta ^{n_{+}} \xrightarrow { \tau _{-} \times \tau _{+} } S^{-} \times S^{+}.$

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

Exercise 1.1.3.7. Show that the inequality of Proposition 1.1.3.6 is sharp. That is, if $S^{-}$ and $S^{+}$ are nonempty simplicial sets of dimensions $k_{-}$ and $k_{+}$, respectively, then the product $S^{-} \times S^{+}$ has dimension $k_{-} + k_{+}$.

We next show that, if $S$ is a simplicial set of dimension $\leq k$, then it can be recovered from its $n$-simplices for $n \leq k$ (Proposition 1.1.3.11). Our proof will make use of the following:

Proposition 1.1.3.8. Let $\sigma : \Delta ^{n} \rightarrow S$ be a morphism of simplicial sets. Then $\sigma$ can be factored as a composition

$\Delta ^{n} \xrightarrow {\alpha } \Delta ^{m} \xrightarrow { \tau } S,$

where $\alpha$ corresponds to a surjective map of linearly ordered sets $[n] \twoheadrightarrow [m]$ and $\tau$ is a nondegenerate $m$-simplex of $S$. Moreover, this factorization is unique.

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

$\tau = \tau \circ \alpha \circ \beta = \sigma \circ \beta = \tau ' \circ \alpha ' \circ \beta .$

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

$\tau ' = \tau ' \circ \alpha ' \circ \beta ' = \sigma \circ \beta ' = \tau \circ \alpha \circ \beta ' = \tau \circ \alpha '' \circ \alpha ' \circ \beta ' = \tau \circ \alpha ''.$

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 $\tau = \tau '$. $\square$

Construction 1.1.3.9 (The Category of Simplices). Let $S_{\bullet }$ be a simplicial set. We define a category $\operatorname{{\bf \Delta }}_{S}$ as follows:

• The objects of $\operatorname{{\bf \Delta }}_{S}$ are pairs $([n], \sigma )$, where $[n]$ is an object of $\operatorname{{\bf \Delta }}$ and $\sigma$ is an $n$-simplex of $S$.

• A morphism from $([n], \sigma )$ to $([n'], \sigma ')$ in the category $\operatorname{{\bf \Delta }}_{S}$ is a nondecreasing function $f: [n] \rightarrow [n']$ with the property that the induced map $S_{n'} \rightarrow S_{n}$ carries $\sigma '$ to $\sigma$.

We will refer to $\operatorname{{\bf \Delta }}_{S}$ as the category of simplices of $S$. If $k$ is an integer, we let $\operatorname{{\bf \Delta }}_{S, \leq k}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{S}$ spanned by those objects $([n], \sigma )$ satisfying $n \leq k$.

Remark 1.1.3.10. Passage from a simplicial set $S$ to the category of simplices $\operatorname{{\bf \Delta }}_{S}$ is a special case of the category of elements construction (see Variant 5.2.6.2), which we will return to in ยง5.2.6.

Proposition 1.1.3.11. Let $k$ be an integer and let $S$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S$ has dimension $\leq k$.

$(2)$

The simplicial set $S$ can be realized as the colimit of a diagram $\varinjlim _{J \in \operatorname{\mathcal{J}}} S(J)$, where each $S(J)$ has dimension $\leq k$.

$(3)$

The simplicial set $S$ can be realized as the colimit of a diagram $\varinjlim _{J \in \operatorname{\mathcal{J}}} S(J)$, where each $S(J)$ is a standard simplex of dimension $\leq k$.

$(4)$

The tautological map

$\varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S, \leq k} } \Delta ^ n \rightarrow S$

is an isomorphism of simplicial sets.

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

$\Delta ^{n} \xrightarrow {\gamma } \Delta ^{p} \xrightarrow { \rho } S,$

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

$\sigma \circ \widetilde{\tau } = f( \tau ) = f( \tau ' ) = \sigma ' \circ \widetilde{\tau }'.$

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 $T$. $\square$

Remark 1.1.3.12. Proposition 1.1.3.11 can be reformulated using the language of Kan extensions (see Definition 7.3.0.1): it asserts that a simplicial set $S: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ has dimension $\leq k$ if and only if it is left Kan extended from the full subcategory of $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ spanned by the objects $\{ [n] \} _{n \leq k}$.

Remark 1.1.3.13. It follows from the proof of Proposition 1.1.3.11 that every simplicial set $S$ can be recovered as the colimit $\varinjlim _{ ( [n], \sigma ) \in \operatorname{{\bf \Delta }}_{S} } \Delta ^ n$. In fact, this is a general feature of presheaf categories: see Theorem 8.4.2.1 for an $\infty$-categorical counterpart.

Corollary 1.1.3.14. Let $k$ be an integer and let $f_{\bullet }: S_{\bullet } \rightarrow T_{\bullet }$ be a morphism between simplicial sets having dimension $\leq k$. Suppose that, for every nonnegative integer $n \leq k$, the map of sets $f_{n}: S_{n} \rightarrow T_{n}$ is a bijection. Then $f$ is an isomorphism of simplicial sets.