1.4.3 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 1.4.3.1. 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 source and target. A homotopy from $f$ to $g$ is a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d^{2}_0( \sigma ) = \operatorname{id}_{D}$, $d^{2}_1(\sigma ) = g$, and $d^{2}_2(\sigma ) = f$, as depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & D \ar [dr]^{\operatorname{id}_ D} & \\ C \ar [ur]^{f} \ar [rr]^{g} & & D. } \]
We will say that $f$ and $g$ are homotopic if there exists a homotopy from $f$ to $g$.
Example 1.4.3.2. 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 source and target) are homotopic as morphisms of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ if and only if $f=g$.
Example 1.4.3.3. 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 $\operatorname{Sing}_{\bullet }(X)$ (in the sense of Definition 1.4.3.1) if and only if 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 1.4.3.4 for a more precise statement).
Exercise 1.4.3.4. Let $\pi : [0,1] \times [0,1] \rightarrow | \Delta ^2 |$ denote the continuous function given by the formula $\pi (s,t) = ( 1-s, (1-t)s, ts )$. 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 1.4.3.1) to the set of continuous functions $H$ satisfying the requirements of Example 1.4.3.3.
Proposition 1.4.3.5. 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 $s^{1}_1(f)$ 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 1.4.3.5. 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 $h$ and $f$ to $g$, respectively, and let $\sigma _0$ be the $2$-simplex given by the constant map $\Delta ^{2} \rightarrow \Delta ^{0} \xrightarrow {Y} \operatorname{\mathcal{C}}$. 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 Proposition 1.2.4.7), 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 $d^{2}_1(\tau )$ as a homotopy from $g$ to $h$.
$\square$
Note that there is a potential asymmetry in Definition 1.4.3.1: 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 identical 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 1.4.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f,g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$ having the same source and target. 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 $d^{2}_0( \sigma ) = \operatorname{id}_{Y}$, $d^{2}_1(\sigma ) = g$, and $d^{2}_2(\sigma ) = f$, as depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{\operatorname{id}_ Y} & \\ X \ar [ur]^{f} \ar [rr]^{g} & & Y. } \]
- $(2)$
There exists a $2$-simplex $\tau $ of $\operatorname{\mathcal{C}}$ satisfying $d^{2}_0( \tau ) = f$, $d^{2}_1(\tau ) = g$, and $d^{2}_2(\tau ) = \operatorname{id}_ X$, as depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{f} & \\ X \ar [ur]^{\operatorname{id}_ X} \ar [rr]^{g} & & Y. } \]
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 1.4.3.5), 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 $( \sigma , s^{1}_1(g), \bullet , s^{1}_0(g) )$ as a map of simplicial sets $\rho _0: \Lambda ^{3}_{2} \rightarrow \operatorname{\mathcal{C}}$ (see Proposition 1.2.4.7), 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 $\tau = d^{3}_2(\rho )$ has the properties required by $(2)$.
$\square$
Using Proposition 1.4.3.6, we can formulate the notion of homotopy in a more symmetric form:
Corollary 1.4.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f,g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$ having the same source and target. Then $f$ and $g$ are homotopic (in the sense of Definition 1.4.3.1) if and only if there exists a map of simplicial sets $H: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ satisfying $H|_{ \{ 0\} \times \Delta ^1} = f$, $H|_{ \{ 1\} \times \Delta ^1} = g$, $H|_{ \Delta ^1 \times \{ 0\} } = \operatorname{id}_ X$, and $H|_{ \Delta ^1 \times \{ 1\} } = \operatorname{id}_{Y}$, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rrrr]^{f} \ar [dddd]^{\operatorname{id}_ X} \ar [ddddrrrr]^{h} & & & & Y \ar [dddd]^{\operatorname{id}_ Y} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ X \ar [rrrr]^{g} & & & & Y. } \]
Proof.
The “only if” direction is clear: if $\sigma $ is a homotopy from $f$ to $g$ (in the sense of Definition 1.4.3.1), then we can extend $\sigma $ to a map $H: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ by taking $\tau $ to be the degenerate simplex $s^{1}_0( g )$. Conversely, suppose that there exists a map $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rrrr]^{f} \ar [dddd]^{\operatorname{id}_ X} \ar [ddddrrrr]^{h} & & & & Y \ar [dddd]^{\operatorname{id}_ Y} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ X \ar [rrrr]^{g} & & & & Y. } \]
Then the $2$-simplex $\sigma $ is a homotopy from $f$ to $h$, and the $2$-simplex $\tau $ guarantees that $g$ is homotopic to $h$ (by virtue of Proposition 1.4.3.6). Since homotopy is an equivalence relation (Proposition 1.4.3.5), it follows that $f$ is homotopic to $g$.
$\square$