Kerodon

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

1.3.3 Homotopies of Morphisms

For any topological space $X$, we can view the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ as an $\infty $-category, where a morphism from a point $x \in X$ to a point $y \in X$ is given by a continuous path $f: [0,1] \rightarrow X$ satisfying $f(0) = x$ and $f(1) = y$. For many purposes (for example, in the study of the fundamental group $\pi _1(X,x)$), it is useful to work not with paths but with homotopy classes of paths (having fixed endpoints). This notion can be generalized to an arbitrary $\infty $-category:

Definition 1.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f,g: C \rightarrow D$ be a pair of morphisms in $\operatorname{\mathcal{C}}$ having the same domain and codomain. A homotopy from $f$ to $g$ is a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d_0( \sigma ) = \operatorname{id}_{D}$, $d_1(\sigma ) = g$, and $d_2(\sigma ) = f$, as depicted in the diagram

\[ \xymatrix { & D \ar [dr]^{\operatorname{id}_ D} & \\ C \ar [ur]^{f} \ar [rr]^{g} & & C. } \]

We will say that $f$ and $g$ are homotopic if there exists a homotopy from $f$ to $g$.

Example 1.3.3.2. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then a pair of morphisms $f,g: C \rightarrow D$ in $\operatorname{\mathcal{C}}$ (having the same domain and codomain) are homotopic as morphisms of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ if and only if $f=g$.

Example 1.3.3.3. Let $X$ be a topological space. Suppose we are given points $x,y \in X$ and a pair of continuous paths $f,g: [0,1] \rightarrow X$ satisfying $f(0) = x = g(0)$ and $f(1)= y = g(1)$. Then $f$ and $g$ are homotopic as morphisms of the $\infty $-category $\operatorname{Sing}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 1.3.3.1 if and only the paths $f$ and $g$ are homotopic relative to their endpoints: that is, if and only if there exists a continuous function $H: [0,1] \times [0,1] \rightarrow X$ satisfying

\[ H(s,0) = f(s) \quad H(s,1) = g(s) \quad H(0, t) = x \quad H(1,t) = y \]

(see Exercise 1.3.3.4 for a more precise statement).

Exercise 1.3.3.4. Let $\pi : [0,1] \times [0,1] \rightarrow | \Delta ^2 |$ denote the continuous function given by the formula $\pi (s,t) = ( 1-s, ts, (1-t)s )$. For any topological space $X$, the construction $\sigma \mapsto \sigma \circ \pi $ determines a map from the set $\operatorname{Sing}_{2}(X)$ of singular $2$-simplices of $X$ to the set of all continuous functions $H: [0,1] \times [0,1] \rightarrow X$. Show that, if $f,g: [0,1] \rightarrow X$ are continuous paths satisfying $f(0) = g(0)$ and $f(1) = g(1)$, then the construction $\sigma \mapsto \sigma \circ \pi $ induces a bijection from the set of homotopies from $f$ to $g$ (in the sense of Definition 1.3.3.1) to the set of continuous functions $H$ satisfying the requirements of Example 1.3.3.3.

Proposition 1.3.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X,Y \in \operatorname{\mathcal{C}}$, and let $E$ denote the collection of all morphisms from $X$ to $Y$ in $\operatorname{\mathcal{C}}$. Then homotopy is an equivalence relation on $E$.

Proof. We first observe that for any morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the degenerate $2$-simplex $s_1(f)$ is a homotopy from $f$ to itself. It follows that homotopy is a reflexive relation on $E$. We will complete the proof by establishing the following:

$(\ast )$

Let $f,g,h: X \rightarrow Y$ be three morphisms from $X$ to $Y$. If $f$ is homotopic to $g$ and $f$ is homotopic to $h$, then $g$ is homotopic to $h$.

Let us first observe that assertion $(\ast )$ implies Proposition 1.3.3.5. Note that in the special case $f = h$, $(\ast )$ asserts that if $f$ is homotopic to $g$, then $g$ is homotopic to $f$ (since $f$ is always homotopic to itself). That is, the relation of homotopy is symmetric. We can therefore replace the hypothesis that $f$ is homotopic to $g$ in assertion $(\ast )$ by the hypothesis that $g$ is homotopic to $f$, so that $(\ast )$ is equivalent to the transitivity of the relation of homotopy.

It remains to prove $(\ast )$. Let $\sigma _2$ and $\sigma _3$ be $2$-simplices of $\operatorname{\mathcal{C}}$ which are homotopies from $f$ to $g$ and $f$ to $h$, respectively, and let $\sigma _0$ be the $2$-simplex given by the constant map $\Delta ^{2} \rightarrow \Delta ^{0} \xrightarrow { X} \operatorname{\mathcal{C}}$. Then the tuple $( \sigma _0, \bullet , \sigma _2, \sigma _3)$ determines a map of simplicial sets $\tau _0: \Lambda ^3_1 \rightarrow \operatorname{\mathcal{C}}$ (see Exercise 1.1.2.14), depicted informally by the diagram

\[ \xymatrix@C =70pt@R=70pt{ & Y \ar [r]^{\operatorname{id}_ Y} \ar [drr]^{\operatorname{id}_ Y} & Y \ar@ {-->}[dr]^{\operatorname{id}_ Y}& \\ X \ar [ur]^{f} \ar@ {-->}[urr]^{g} \ar@ {-->}[rrr]^{h} & & & Y; } \]

here the dotted arrows represent the boundary of the “missing” face of the horn $\Lambda ^{3}_1$. Our hypothesis that $\operatorname{\mathcal{C}}$ is an $\infty $-category guarantees that $\tau _0$ can be extended to a $3$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. We can then regard the face $d_1(\tau )$ as a homotopy from $g$ to $h$. $\square$

Note that there is a potential asymmetry in Definition 1.3.3.1: if $f,g: X \rightarrow Y$ are two morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$, then the datum of a homotopy from $f$ to $g$ in the $\infty $-category $\operatorname{\mathcal{C}}$ is not equivalent to the datum of a homotopy from $f$ to $g$ in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Nevertheless, we have the following:

Proposition 1.3.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f,g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$ having the same domain and codomain. Then $f$ and $g$ are homotopic if and only if they are homotopic when regarded as morphisms of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. In other words, the following conditions are equivalent:

$(1)$

There exists a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d_0( \sigma ) = \operatorname{id}_{D}$, $d_1(\sigma ) = g$, and $d_2(\sigma ) = f$, as depicted in the diagram

\[ \xymatrix { & D \ar [dr]^{\operatorname{id}_ D} & \\ C \ar [ur]^{f} \ar [rr]^{g} & & C. } \]
$(2)$

There exists a $2$-simplex $\tau $ of $\operatorname{\mathcal{C}}$ satisfying $d_0( \tau ) = f$, $d_1(\tau ) = g$, and $d_2(\tau ) = \operatorname{id}_ X$, as depicted in the diagram

\[ \xymatrix { & C \ar [dr]^{f} & \\ C \ar [ur]^{\operatorname{id}_ C} \ar [rr]^{g} & & D. } \]

Proof. We will show that $(1)$ implies $(2)$; the proof of the reverse implication is similar. Assume that $f$ is homotopic to $g$. Since the relation of homotopy is symmetric (Proposition 1.3.3.5), it follows that $g$ is also homotopic to $f$. Let $\sigma $ be a homotopy from $g$ to $f$. Then we can regard the tuple of $2$-simplices $( \sigma , s_1(g), \bullet , s_0(g) )$ as a map of simplicial sets $\rho _0: \Lambda ^{3}_{2} \rightarrow \operatorname{\mathcal{C}}$ (see Exercise 1.1.2.14), depicted informally in the diagram

\[ \xymatrix@C =70pt@R=70pt{ & X \ar [r]^{g} \ar@ {-->}[drr]^{f} & Y \ar [dr]^{\operatorname{id}_ Y}& \\ X \ar@ {-->}[ur]^{\operatorname{id}_ X} \ar [urr]^{g} \ar@ {-->}[rrr]^{g} & & & Y, } \]

where the dotted arrows indicate the boundary of the “missing” face of the horn $\Lambda ^{3}_{2}$. Using our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can extend $\rho _0$ to a $3$-simplex $\rho $ of $\operatorname{\mathcal{C}}$. Then the face $\tau = d_2(\rho )$ has the properties required by $(2)$. $\square$