# Kerodon

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### 3.2.4 Connectivity and Contractibility

We now study the class of contractible simplicial sets.

Definition 3.2.4.1. Let $X$ be a simplicial set. We say that $X$ is contractible if the projection map $X \rightarrow \Delta ^0$ is a homotopy equivalence (Definition 3.1.6.1).

Example 3.2.4.2. Let $\operatorname{\mathcal{C}}$ be a category. If $\operatorname{\mathcal{C}}$ has an initial object or a final object, then the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is contractible (this is a special case of Proposition 3.1.6.9). In particular, for every integer $n \geq 0$, the standard simplex $\Delta ^ n$ is contractible.

Though the condition of contractibility makes sense for any simplicial set $X$, we will be primarily interested in the special case where $X$ is a Kan complex. In this case, Definition 3.2.4.1 agrees with Definition 1.4.5.8:

Theorem 3.2.4.3. Let $X$ be a Kan complex. The following conditions are equivalent:

$(1)$

The Kan complex $X$ is contractible.

$(2)$

The Kan complex $X$ is connected and the homotopy groups $\pi _{n}(X,x)$ vanish for each $n > 0$ and every choice of base point $x \in X$.

$(3)$

The projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration of simplicial sets.

Remark 3.2.4.4. In the formulation of Theorem 3.2.4.3, we can replace $(2)$ by the following a priori weaker condition:

$(2')$

The Kan complex $X$ is connected and, for some choice of base point $x \in X$, the homotopy groups $\pi _{n}(X,x)$ vanish for each $n > 0$.

See Example 3.2.2.17.

We will deduce Theorem 3.2.4.3 from a more general result, which does not require the vanishing of all the homotopy groups of $X$. First, we need to introduce some terminology.

Definition 3.2.4.5. Let $X$ be a Kan complex and let $n$ be a nonnegative integer. We say that $X$ is $n$-connective if it is nonempty and, for every vertex $x \in X$ and every integer $0 \leq m < n$, the set $\pi _{m}(X,x)$ consists of a single element.

Warning 3.2.4.6. The terminology of Definition 3.2.4.5 is not standard. Many authors refer to a Kan complex $X$ as $n$-connected if it is $(n+1)$-connective in the sense of Definition 3.2.4.5.

Remark 3.2.4.7. Let $X$ be a Kan complex. It follows from Example 3.2.2.17 that the isomorphism class of the homotopy group $\pi _{m}(X,x)$ depends only on the connected component $[x] \in \pi _0(X)$. Consequently, to verify that $X$ is $n$-connective, it suffices to that the sets $\pi _{m}(X,x)$ vanish for $0 \leq m < n$ for some choice of vertex $x \in X$ (when $m = 0$, this condition guarantees that any other vertex $y \in X$ belongs to the same connected component).

Variant 3.2.4.9. Let $X$ be a simplicial set and let $n \geq 0$ be an integer. Using Corollary 3.1.7.2, we can choose an anodyne map $X \hookrightarrow Q$, where $Q$ is a Kan complex. We will say that $X$ is $n$-connective if the Kan complex $Q$ is $n$-connective, in the sense of Definition 3.2.4.5. By virtue of Remark 3.2.4.8 (and Warning 3.1.7.3), this condition is independent of the choice of $Q$.

Example 3.2.4.10. A simplicial set $X$ is $0$-connective if and only if it is nonempty.

Example 3.2.4.12. A Kan complex $X$ is $2$-connective if and only if it is simply connected: that is, $X$ is connected and the fundamental group $\pi _{1}(X,x)$ vanishes (by virtue of Remark 3.2.4.7, this condition does not depend on the choice of base point $x \in X$).

Our next goal is to show that the Definition 3.2.4.5 can be reformulated as a lifting property (Proposition 3.2.4.18).

Definition 3.2.4.13. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. We will say that $f$ is nullhomotopic if there exists a vertex $y \in Y$ for which $f$ is homotopic to the constant morphism $X \rightarrow \{ y\} \hookrightarrow Y$.

Example 3.2.4.14. Let $X$ be a simplicial set, and let $\emptyset$ denote the empty simplicial set. Then there is a unique morphism of simplicial sets $\emptyset \hookrightarrow X$, which is nullhomotopic if and only if $X$ is nonempty (note that, by the convention of Definition 3.2.4.13, the identity map $\emptyset \rightarrow \emptyset$ is not considered to be nullhomotopic).

Example 3.2.4.15. Let $(X,x)$ be a pointed Kan complex, let $n > 0$ be a positive integer, and let $\sigma : \Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow (X,x)$ be a morphism of pointed simplicial sets. Then $\sigma$ is nullhomotopic (in the sense of Definition 3.2.4.13) if and only if the pointed homotopy class $[\sigma ]$ is equal to the identity element in the homotopy group $\pi _{n}(X,x)$.

Remark 3.2.4.16. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of simplicial sets. If either $f$ or $g$ is nullhomotopic, then the composition $g \circ f$ is nullhomotopic.

Lemma 3.2.4.17. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. Then a morphism of simplicial sets $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow X$ is nullhomotopic if and only if it can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$.

Proof. Suppose first that $\sigma _0$ is homotopic to a constant map $\sigma '_{0}: \operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. Since $\sigma '_{0}$ can be extended to a map $\sigma ': \Delta ^ n \rightarrow \{ x\} \hookrightarrow X$, it follows from the homotopy extension lifting property (Remark 3.1.5.3) that $\sigma _0$ can be extended to an $n$-simplex of $X$.

For the converse, assume that $\sigma _0$ admits an extension $\sigma : \Delta ^ n \rightarrow X$. The projection map $\Delta ^ n \rightarrow \Delta ^0$ is a homotopy equivalence (Proposition 3.1.6.9). It follows that the induced map $X \simeq \operatorname{Fun}( \Delta ^0, X) \rightarrow \operatorname{Fun}( \Delta ^ n, X)$ is also a homotopy equivalence. In particular, it induces a surjection on connected components, so $\sigma$ is nullhomotopic. Applying Remark 3.2.4.16, we deduce that $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^ n }$ is also nullhomotopic. $\square$

Proposition 3.2.4.18. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. The following conditions are equivalent:

$(1)$

The Kan complex $X$ is $n$-connective.

$(2)$

Let $B$ be a simplicial set of dimension $\leq n$ and let $A \subseteq B$ be a simplicial subset. Then every morphism $f_0: A \rightarrow X$ admits an extension $f: B \rightarrow X$.

$(3)$

Let $A$ be a simplicial set of dimension $< n$. Then every morphism $f: A \rightarrow X$ is nullhomotopic.

$(4)$

For every integer $0 \leq m \leq n$, every morphism $\operatorname{\partial \Delta }^{m} \rightarrow X$ can be extended to an $m$-simplex of $X$.

Proof. Our proof proceeds by induction on $n$. We first show that $(2)$ implies $(3)$. Applying assumption $(2)$ to the inclusion map $\emptyset \subseteq \Delta ^0$, we deduce that there exists a vertex $x \in X$. It will therefore suffice to show that if $A$ is a simplicial set of dimension $< n$, then every pair of morphisms $f_0, f_1: A \rightarrow X$ are homotopic (in particular, $f_0$ is homotopic to the constant map $A \rightarrow \{ x\}$). This follows by applying $(2)$ to the inclusion map $\operatorname{\partial \Delta }^1 \times A \hookrightarrow \Delta ^1 \times A$ (see Proposition 1.1.3.11).

The implication $(3) \Rightarrow (4)$ follows from Lemma 3.2.4.17. We next show that $(4)$ implies $(1)$. Applying assumption $(4)$ in the case $m = 0$, we conclude that $X$ is nonempty (Example 3.2.4.14). Fix a vertex $x \in X$; we wish to show that every element $\eta \in \pi _{k}(X,x)$ is trivial for $0 \leq k < n$. Choose a $k$-simplex $\sigma : \Delta ^{k} \rightarrow X$ which represents $\eta$, so that $\sigma |_{ \operatorname{\partial \Delta }^ k }$ is the constant morphism taking the value $x$ Then $\sigma$ can be extended to a map $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ for which $\tau _0 |_{ \Lambda ^{k+1}_{0} }$ is the constant morphism taking the value $x$. Assumption $(4)$ guarantees that $\tau _0$ is can be extended to an $(k+1)$-simplex $\tau$ of $X$ (Lemma 3.2.4.17). Let $u: \Delta ^1 \times \Delta ^ k \rightarrow \Delta ^{k+1}$ be the morphism given on vertices by the formula

$u(i,j) = \begin{cases} j & \text{ if i=0 } \\ k+1 & \text{ if i=1. } \end{cases}$

Then the composition $\Delta ^1 \times \Delta ^{k} \xrightarrow {\rho } \Delta ^{k+1} \xrightarrow {\tau } X$ determines a homotopy from $\sigma$ to the constant map $\Delta ^{k} \rightarrow \{ x\} \hookrightarrow X$, which is constant along the boundary $\operatorname{\partial \Delta }^{k}$.

We now complete the proof by showing that $(1)$ implies $(2)$. Assume that $X$ is $n$-connective, let $B$ be a simplicial set of dimension $\leq n$, let $A \subseteq B$ be a simplicial subset, and suppose we are given a morphism $f_0: A \rightarrow X$; we wish to show that $f_0$ admits an extension $B \rightarrow X$. Using Proposition 1.1.3.13, we can reduce to the case where $B = \Delta ^{m}$ is a standard simplex of dimension $m \leq n$ and $A = \operatorname{\partial \Delta }^{m}$ is its boundary. Since $X$ is nonempty, we may assume that $m > 0$. If $m = 1$, then $X$ is connected (Example 3.2.4.11), and the desired result follows from Proposition 1.1.9.10. We may therefore assume without loss of generality that $m \geq 2$. Let $K$ be the $(m-2)$-skeleton of $\Delta ^{m}$. Since $X$ is $(n-1)$-connective, our inductive hypothesis guarantees that $f_0|_{ K }$ is nullhomotopic. Using Remark 3.1.5.3, we can choose a homotopy $\alpha _0$ from $f_0$ to a morphism $g_0: \operatorname{\partial \Delta }^{m} \rightarrow X$ such that $g_0|_{ K }$ is the constant map taking some value $x \in X$. Since $m \leq n$, our assumption that $X$ is $n$-connective guarantees that the homotopy group $\pi _{m-1}(X,x)$ is trivial. Using the characterization of the group structure on $\pi _{m-1}(X,x)$ given in Theorem 3.2.2.10, we conclude that $g_0$ admits an extension $g: \Delta ^{m} \rightarrow X$. Using Remark 3.1.5.3 again, we can extend $\alpha _0$ to a homotopy from $f$ to $g$, where $f: \Delta ^{m} \rightarrow X$ is some extension of $f_0$. $\square$

Proof of Theorem 3.2.4.3. Let $X$ be a Kan complex. We wish to show that the following conditions are equivalent:

$(1)$

The Kan complex $X$ is contractible: that is, the projection map $X \rightarrow \Delta ^0$ is a homotopy equivalence.

$(2)$

The Kan complex $X$ is $n$-connective for every integer $n \geq 0$.

$(3)$

The projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration: that is, every morphism of simplicial sets $\operatorname{\partial \Delta }^ n \rightarrow X$ can be extended to an $n$-simplex of $X$.

The implication $(1) \Rightarrow (2)$ follows from Remark 3.2.2.16, and the implication $(3) \Rightarrow (1)$ is a special case of Proposition 3.1.6.10. The equivalence of $(2)$ and $(3)$ follows from Proposition 3.2.4.18. $\square$

When working with simplicial sets which are not Kan complexes, it will generally be convenient to work with the following variant of Definition 3.2.4.1.

Definition 3.2.4.19. Let $X$ be a simplicial set. We say that $X$ is weakly contractible if the projection map $X \rightarrow \Delta ^{0}$ is a weak homotopy equivalence (Definition 3.1.6.12).

Remark 3.2.4.20. Let $X$ be a simplicial set. If $X$ is contractible, then it is weakly contractible. The converse holds if $X$ is a Kan complex (Proposition 3.1.6.13). Beware that the converse is false in general (Exercise 3.1.6.21).

Remark 3.2.4.21. Let $f: X \rightarrow Y$ be a weak homotopy equivalence of simplicial sets. Then $X$ is weakly contractible if and only if $Y$ is weakly contractible (see Remark 3.1.6.16). If $f$ is a homotopy equivalence, then $X$ is contractible if and only if $Y$ is contractible (see Remark 3.1.6.7).

Example 3.2.4.22. Let $n$ be a positive integer. For $0 \leq i \leq n$, the horn $\Lambda ^{n}_{i}$ is weakly contractible. This follows from Remark 3.2.4.21, since the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a weak homotopy equivalence (Proposition 3.1.6.14) and the simplex $\Delta ^ n$ is contractible (Example 3.2.4.2).

Proposition 3.2.4.23. Let $X$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $X$ is weakly contractible (in the sense of Definition 3.2.4.19).

$(2)$

For every integer $n \geq 0$, $X$ is $n$-connective (in the sense of Variant 3.2.4.9).

$(3)$

For every Kan complex $Y$, every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic.

Proof. We first show that $(1) \Leftrightarrow (2)$. Using Corollary 3.1.7.2, we can choose an anodyne morphism $X \hookrightarrow X'$, where $X'$ is a Kan complex. By virtue of Remark 3.2.4.21, we can replace $X$ by $X'$ and thereby reduce to proving Proposition 3.2.4.23 under the assumption that $X$ is a Kan complex. In this case, $X$ is weakly contractible if and only if it is contractible (Remark 3.2.4.20). The desired result now follows from Proposition 3.2.4.18.

We now show that $(1) \Leftrightarrow (3)$. Without loss of generality, we may assume that $X$ is nonempty (note that if $X$ is empty, then $X$ is a Kan complex but the identity map $\operatorname{id}_{X}: X \rightarrow X$ is not nullhomotopic). Fix a vertex $x \in X$. By definition, $X$ is weakly contractible if and only if, for every Kan complex $Y$, the diagonal map $\delta : X \rightarrow \operatorname{Fun}(X,Y)$ induces a bijection on connected components. Note that $\delta$ admits a left inverse (given by the evaluation map $\operatorname{Fun}(X,X) \rightarrow \operatorname{Fun}( \{ x\} , X) \simeq Y$), and is therefore automatically injective on connected components. Consequently, $X$ is weakly contractible if and only if, for every Kan complex $Y$, the map $\delta$ is surjective at the level of connected components: that is, if and only if every morphism $f: X \rightarrow Y$ is homotopic to a constant map. $\square$