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).
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3.2.4 Contractibility
We now study the class of contractible simplicial sets.
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 1.5.5.8:
Theorem 3.2.4.3. Let $X$ be a Kan complex. The following conditions are equivalent:
The Kan complex $X$ is contractible.
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 $x \in X$.
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:
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 3.2.2.18.
For the proof of Theorem 3.2.4.3, it will be convenient to introduce some terminology.
Definition 3.2.4.5. 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 3.2.4.6. 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 3.2.4.5, the identity map $\emptyset \rightarrow \emptyset $ is not considered to be nullhomotopic).
Example 3.2.4.7. 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 3.2.4.5) if and only if the pointed homotopy class $[\sigma ]$ is equal to the identity element in the homotopy group $\pi _{n}(X,x)$. See Proposition 3.2.1.15.
Exercise 3.2.4.8. Let $X$ be a simplicial set. Show that the following conditions are equivalent:
The simplicial set $X$ is contractible: that is, the projection map $X \rightarrow \Delta ^0$ is a homotopy equivalence (Definition 3.2.4.1).
The identity morphism $\operatorname{id}_{X}: X \rightarrow X$ is nullhomotopic.
Every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic.
Every morphism of simplicial sets $g: Z \rightarrow X$ is nullhomotopic.
In particular, these conditions are satisfied in the special case where $X = \Delta ^{n}$ is a standard simplex.
Remark 3.2.4.9. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, and let $f': X \rightarrow Y$ be a morphism which is homotopic to $f$. Then $f$ is nullhomotopic if and only if $f'$ is nullhomotopic.
Remark 3.2.4.10. 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.
Remark 3.2.4.11. Let $X$ be a simplicial set. Then $X$ is weakly contractible if and only if, for every Kan complex $Y$, every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic. To prove this, we may assume that $X$ is nonempty (otherwise the identity morphism $\operatorname{id}_{X}$ is not nullhomotopic; see Example 3.2.4.6). Then, for any Kan complex $Y$, the diagonal map $\delta _{Y}: X \rightarrow \operatorname{Fun}(X,Y)$ admits a left inverse (given by evaluation at any vertex $x \in X$), and is automatically injective on connected components. It follows that $X$ is weakly contractible if and only if, for every Kan complex $Y$, the morphism $\delta _{Y}$ is also surjective on connected components: that is, every morphism $f: X \rightarrow Y$ is homotopic to a constant map.
Lemma 3.2.4.12. Let $X$ be a Kan complex and let $0 \leq i \leq n$ be integers with $n > 0$. Then every morphism $\sigma _0: \Lambda ^{n}_{i} \rightarrow X$ is nullhomotopic.
Proof. Since $X$ is a Kan complex, we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$. By virtue of Remark 3.2.4.10, it will suffice to show that $\sigma $ is nullhomotopic, which is a special case of Exercise 3.2.4.8. $\square$
Variant 3.2.4.13. 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 of $X$. The “if” direction follows immediately from Exercise 3.2.4.8 (and does not require the assumption that $X$ is a Kan complex). For the converse, suppose 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 also be extended to an $n$-simplex of $X$.
Lemma 3.2.4.14. Let $X$ be a Kan complex and let $n \geq 2$ be an integer. The following conditions are equivalent:
Every morphism $\operatorname{\partial \Delta }^{n} \rightarrow X$ can be extended to an $n$-simplex of $X$ (that is, it is nullhomotopic).
For every vertex $x \in X$, the homotopy group $\pi _{n-1}(X,x)$ is trivial.
Proof. We first show that $(a_ n)$ implies $(b_ n)$. Fix a vertex $x \in X$ and an $(n-1)$-simplex $\sigma : \Delta ^{n-1} \rightarrow X$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n-1}}$ is the constant map taking the value $x$. Amalgamating $\sigma $ with the constant map $\Lambda ^{n}_{n} \rightarrow X$, we obtain a morphism $\tau : \operatorname{\partial \Delta }^{n} \rightarrow X$. If condition $(1)$ is satisfied, then $\tau $ can be extended to an $n$-simplex of $X$. Theorem 3.2.2.10 then guarantees that the (pointed) homotopy class $[\sigma ]$ is the identity element of the group $\pi _{n-1}(X,x)$.
We now show that $(b_ n)$ implies $(a_ n)$. Let $\tau : \operatorname{\partial \Delta }^{n} \rightarrow X$ be any morphism of simplicial sets. Using Lemma 3.2.4.12, we see that the restriction $\tau |_{ \Lambda ^{n}_{n} }$ is nullhomotopic. Applying the homotopy lifting property (Remark 3.1.5.3), we conclude that $\tau $ is homotopic to a morphism $\tau ': \operatorname{\partial \Delta }^{n} \rightarrow X$ for which $\tau '|_{ \Lambda ^{n}_{n} }$ is the constant map taking the value $x$, for some vertex $x \in X$. In particular, $\tau '$ is constant when restricted to the $(n-2)$-skeleton of $\Delta ^{n}$. If the homotopy group $\pi _{n-1}(X,x)$ is trivial, then Theorem 3.2.2.10 guarantees that $\tau '$ can be extended to an $n$-simplex of $X$. Applying Remark 3.1.5.3 again, we conclude that $\tau $ can also be extended to an $n$-simplex of $X$. $\square$
Variant 3.2.4.15. In the situation of Lemma 3.2.4.14, the extension condition $(a_ n)$ also makes sense for $n=0$ and $n=1$. Here condition $(a_0)$ is equivalent to the requirement that $\pi _0(X)$ has at least one element (that is, $X$ is nonempty), and condition $(a_1)$ is equivalent to the requirement that $\pi _0(X)$ has at most one element (that is, $X$ is either empty or connected). In particular, $X$ satisfies conditions $(a_0)$ and $(a_1)$ if and only if it is connected.
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 connected and the homotopy groups $\pi _{n}(X,x)$ vanish for every integer $n > 0$ and every vertex $x \in X$.
- $(3)$
The projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration: that is, every morphism of simplicial sets $\operatorname{\partial \Delta }^{m} \rightarrow X$ can be extended to an $m$-simplex of $X$.
The implication $(1) \Rightarrow (2)$ follows from Remark 3.2.2.17, and the implication $(3) \Rightarrow (1)$ is a special case of Proposition 3.1.6.10. The equivalence of $(2)$ and $(3)$ follows from Lemma 3.2.4.14 and Variant 3.2.4.15. $\square$
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 3.2.4.16. 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 3.2.4.17. 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 3.2.4.18. 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 3.2.4.19. Let $n$ be a positive integer. For $0 \leq i \leq n$, the horn $\Lambda ^{n}_{i}$ is weakly contractible. This follows from Remark 3.2.4.18, 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).