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1.3.7 Equivalences

Recall that a morphism $f: X \rightarrow Y$ in a category $\operatorname{\mathcal{C}}$ is an isomorphism if there exists a morphism $g: Y \rightarrow X$ satisfying $f \circ g = \operatorname{id}_{Y}$ and $g \circ f = \operatorname{id}_ X$. This notion has an $\infty $-categorical analogue:

Definition 1.3.7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that $f$ is an equivalence if the homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We will say that two objects $X,Y \in \operatorname{\mathcal{C}}$ are equivalent if there exists an equivalence between $X$ to $Y$ (that is, if $X$ and $Y$ are isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$).

Example 1.3.7.2. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if it is an equivalence when regarded as a morphism of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Remark 1.3.7.3. If $f: X \rightarrow Y$ is an equivalence in an $\infty $-category $\operatorname{\mathcal{C}}$, then one should regard the objects $X,Y \in \operatorname{\mathcal{C}}$ as essentially interchangeable, just as isomorphic objects of an ordinary category are essentially interchangeable. Our use of the term “equivalence” rather than “isomorphism” is motivated by the desire to avoid confusion in situations where a class of mathematical objects admits both $1$-categorical and $\infty $-categorical descriptions. For example:

  • The collection of topological spaces can be organized into an ordinary category $\operatorname{Top}$, whose morphisms are continuous functions and whose isomorphisms are homeomorphisms. However, it can also be organized into an $\infty $-category $\operatorname{N}_{\bullet }^{\Delta }( \operatorname{Top})$ (see §) in which the equivalences (in the sense of Definition 1.3.7.1) are homotopy equivalences: that is, continuous functions which admit a homotopy inverse.

  • The collection of (small) categories can be organized into an ordinary category $\operatorname{Cat}$, whose morphisms are functors and whose isomorphisms are functors which are fully faithful and bijective on objects. However, it can also be organized into an $\infty $-category in which the equivalences (in the sense of Definition 1.3.7.1) are equivalences of categories: that is, functors which are fully faithful and bijective on isomorphism classes of objects.

Remark 1.3.7.4 (Two-out-of-three). Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$ and let $h$ be a composition of $f$ and $g$. If any two of the morphisms $f$, $g$, and $h$ is an equivalence, then so is the third.

Definition 1.3.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$. We say that $g$ is a left homotopy inverse of $f$ if the identity morphism $\operatorname{id}_{X}$ is a composition of $f$ and $g$: that is, if we have an equality $[\operatorname{id}_ X] = [g] \circ [f]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We say that $g$ is a right homotopy inverse of $f$ if the identity morphism $\operatorname{id}_{Y}$ is a composition of $g$ and $f$: that is, if we have an equality $[ \operatorname{id}_ Y ] = [f] \circ [g]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We will say that $g$ is a homotopy inverse of $f$ if it is both a left and a right homotopy inverse of $f$.

Remark 1.3.7.6. Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$. Then the condition that $g$ is a left homotopy inverse (right homotopy inverse, homotopy inverse) to $f$ depends only on the homotopy classes $[f]$ and $[g]$.

Remark 1.3.7.7. Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$. Then $g$ is left homotopy inverse to $f$ if and only if $f$ is right homotopy inverse to $g$. Both of these conditions are equivalent to the existence of a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d_0(\sigma ) = g$, $d_1(\sigma ) = \operatorname{id}_{X}$, and $d_2(\sigma ) = f$, as depicted in the diagram

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

Remark 1.3.7.8. Let $f: X \rightarrow Y$ be a morphism in $\infty $-category $\operatorname{\mathcal{C}}$. Suppose that $f$ admits a left homotopy inverse $g$ and a right homotopy inverse $h$. Then $g$ and $h$ are homotopic: this follows from the calculation

\[ [g] = [g] \circ [ \operatorname{id}_ Y ] = [g] \circ ( [f] \circ [h] ) = ( [g] \circ [f] ) \circ [h] = [ \operatorname{id}_ Y ] \circ [h] = [h]. \]

It follows that both $g$ and $h$ are homotopy inverse to $f$.

Remark 1.3.7.9. Let $f: X \rightarrow Y$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$. It follows from Remark 1.3.7.8 that the following conditions are equivalent:

$(1)$

The morphism $f$ is an equivalence.

$(2)$

The morphism $f$ admits a homotopy inverse $g$.

$(3)$

The morphism $f$ admits both left and right homotopy inverses.

In this case, the morphism $g$ is uniquely determined up to homotopy; moreover, any left or right homotopy inverse of $f$ is homotopic to $g$. We will sometimes abuse notation by writing $f^{-1}$ to denote a homotopy inverse to $f$.

Warning 1.3.7.10. Let $f: X \rightarrow Y$ be a morphism in an $\infty $-category $\operatorname{\mathcal{C}}$, and suppose that $g,h: Y \rightarrow X$ are left homotopy inverses to $f$. If $f$ does not admit a right homotopy inverse, then $g$ and $h$ need not be homotopic.

Proposition 1.3.7.11 (Two-out-of-Six). Let $f: W \rightarrow X$, $g: X \rightarrow Y$, and $h: Y \rightarrow Z$ be morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$. If the morphisms $g \circ f$ and $h \circ g$ are equivalences, then $f$, $g$, and $h$ are also equivalences.

Proof. Let $u$ be a homotopy inverse to $g \circ f$. Then the iterated composition $g \circ (f \circ u)$ is homotopic to the identity, so that $g$ admits a right homotopy inverse. Similarly, $g$ admits a left homotopy inverse. It follows that $g$ is an equivalence (Remark 1.3.7.8). Since $f \circ u$ is a right homotopy inverse to $g$, it is homotopy inverse to $g$ (Remark 1.3.7.8), and is therefore also an equivalence. Applying Remark 1.3.7.4, we conclude that $f$ is also an equivalence. A similar argument shows that $h$ is an equivalence. $\square$

Proposition 1.3.7.12. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then every morphism in $\operatorname{\mathcal{C}}$ is an equivalence.

Remark 1.3.7.13. We will see later that the converse to Proposition 1.3.7.12 is also true: if $\operatorname{\mathcal{C}}$ is an $\infty $-category in which every morphism is an equivalence, then $\operatorname{\mathcal{C}}$ is a Kan complex (Theorem ).

Proof of Proposition 1.3.7.12. Let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Then the tuple $(\bullet , \operatorname{id}_{X}, f)$ determines a map of simplicial sets $\sigma _0: \Lambda ^{2}_{0} \rightarrow \operatorname{\mathcal{C}}$ (Exercise 1.1.2.14), which we depict as

\[ \xymatrix { & Y \ar@ {-->}[dr] & \\ X \ar [ur]^{f} \ar [rr]^{\operatorname{id}_ X} & & X. } \]

If $\operatorname{\mathcal{C}}$ is a Kan complex, then we can extend $\sigma _0$ to a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$. Then $\sigma $ exhibits the morphism $g = d_0(\sigma )$ as a left homotopy inverse to $f$. A similar argument shows that $f$ admits a right homotopy inverse, so that $f$ is an equivalence by virtue of Remark 1.3.7.9. $\square$

Example 1.3.7.14. Let $X$ be a topological space. Then every morphism in the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$ is an equivalence. In other words, the category $\pi _{\leq 1}(X) = \mathrm{h} \mathit{ \operatorname{Sing}}_{\bullet }(X)$ is a groupoid. This follows by combining Propositions 1.3.7.12 and 1.1.7.3. However, it is also easy to see directly: if $f$ is a morphism of $\operatorname{Sing}_{\bullet }(X)$, given by a continous path $f: [0,1] \rightarrow X$, then the continuous path

\[ g: [0,1] \rightarrow X \quad \quad g(t) = f(1-t) \]

is a homotopy inverse of $f$ in the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$.