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10.3.2 Quotient Morphisms

Let $X$ and $Y$ be sets, and let $f: X \rightarrow Y$ be a function. The function $f$ determines an equivalence relation $\equiv _{f}$ on $X$, defined by the requirement

\[ (x \equiv _{f} x' ) \Leftrightarrow ( f(x) = f(x') ). \]

If $f$ is surjective, then it induces an bijection from $X / \equiv _{f}$ to $Y$. Stated in more categorical terms, this means that for every set $S$, composition with $f$ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functions $Y \rightarrow S$} \} \ar [d] \\ \{ \textnormal{Functions $g: X \rightarrow S$ satisfying $g(x) = g(x')$ when $f(x) = f(x')$} \} . } \]

Our goal in this section is to study an $\infty $-categorical counterpart of this condition.

Definition 10.3.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve generated by $f$ (see Example 10.3.1.19). We will say that $f$ is a quotient morphism if the composite map

\[ (\operatorname{\mathcal{C}}^{0}_{/Y})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}} \]

is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

Notation 10.3.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ having source $X$ and target $Y$. If $f$ is a quotient morphism, we will often visually emphasize this by denoting $f$ with a double-headed arrow (that is, we will write $f: X \twoheadrightarrow Y$ in place of $f: X \rightarrow Y$). Beware that this notation does not indicate that $f$ is an epimorphism (see Warning 10.3.2.10).

Exercise 10.3.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Show that every isomorphism in $\operatorname{\mathcal{C}}$ is a quotient morphism (see Example 10.3.4.3 for a stronger statement).

Stated more informally, a morphism $f: X \rightarrow Y$ is a quotient morphism if the object $Y$ can be recovered as the colimit $\varinjlim _{C \rightarrow Y} C$, indexed by the $\infty $-category of morphisms $g: C \rightarrow Y$ which factor through $f$. If the $\infty $-category $\operatorname{\mathcal{C}}$ admits pullbacks, this condition admits a more concrete formulation.

Proposition 10.3.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}(X/Y)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ) \rightarrow \operatorname{\mathcal{C}}$ (see Definition 10.2.5.4). Then $f$ is a quotient morphism if and only if $\operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Stated more informally, Proposition 10.3.2.4 asserts that $f: X \rightarrow Y$ is a quotient morphism if and only if it exhibits $Y$ as a geometric realization of the simplicial object depicted in the diagram

\[ \xymatrix@C =50pt@R=50pt{ \cdots \ar@ <.4ex>[r] \ar@ <-.4ex>[r] \ar@ <1.2ex>[r] \ar@ <-1.2ex>[r] & X \times _{Y} X \times _{Y} X \ar@ <.8ex>[r] \ar@ <-.8ex>[r] \ar [r] & X \times _{Y} X \ar@ <.4ex>[r] \ar@ <-.4ex>[r] & X. } \]

The proof will require some preliminaries.

Lemma 10.3.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}(X)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ (see Definition 10.2.4.3). Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$ (Example 10.3.1.19). Then the functor $\operatorname{\check{C}}(X)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}^{0}$ is right cofinal.

Proof. Let $C$ be an object of $\operatorname{\mathcal{C}}$ and let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor corepresented by $C$. Since $h^{C}$ preserves finite products (Proposition 7.4.1.18), the composition

\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { \operatorname{\check{C}}(X)_{\bullet } } \operatorname{\mathcal{C}}\xrightarrow { h^{C} } \operatorname{\mathcal{S}} \]

is a simplicial object of $\operatorname{\mathcal{S}}$ which can be identified with the Čechnerve of the Kan complex $h^{C}( \operatorname{\check{C}}(X)_{0} ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X )$ (Remark 10.2.4.7). If $C$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}$, then the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is nonempty. Applying Corollary 10.2.6.15, we conclude that the geometric realization $| h^{C}( \operatorname{\check{C}}(X)_{\bullet } ) |$ is contractible. The desired result now follows by allowing the object $C$ to vary and applying the criterion of Proposition 7.4.3.11. $\square$

Variant 10.3.2.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}(X/Y)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ) \rightarrow \operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y}$ denote the sieve generated by $f$. Then $\operatorname{\check{C}}(X/Y)_{\bullet }$ determines a right cofinal functor $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}^{0}_{/Y}$.

Proof. Apply Lemma 10.3.2.5 to the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. $\square$

Proof of Proposition 10.3.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}(X/Y)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ) \rightarrow \operatorname{\mathcal{C}}$. We wish to show that $f$ is a quotient morphism if and only if $\operatorname{\check{C}}(X/Y)_{\bullet }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ denote the sieve generated by $X$ and let $Q$ denote the composite map

\[ (\operatorname{\mathcal{C}}^{0}_{/Y})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}. \]

Let us identify $\operatorname{\check{C}}(X/Y)_{\bullet }$ with a functor $F: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}_{/Y}^{\operatorname{op}}$. Unwinding the definitions, we wish to show that $Q$ is a colimit diagram if and only if the composite functor

\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )^{\triangleright } \xrightarrow { F^{\triangleright } } (\operatorname{\mathcal{C}}^{0}_{/Y} )^{\triangleright } \xrightarrow {Q} \operatorname{\mathcal{C}} \]

is a colimit diagram. This is a special case of Corollary 7.2.2.3, since the functor $F$ is right cofinal (Variant 10.3.2.6). $\square$

Corollary 10.3.2.7. Let $\operatorname{\mathcal{C}}$ be a category which admits pullbacks and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ is a quotient morphism in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.3.2.1).

$(2)$

The morphism $f$ is a regular epimorphism: that is, it exhibits $Y$ as a coequalizer of the projection maps $X \times _{Y} X \rightrightarrows X$ (Definition 10.3.0.2).

Example 10.3.2.8. Let $X$ and $Y$ be sets. Then a function $f: X \rightarrow Y$ is a quotient morphism (in the category of sets) if and only if it is surjective (see Example 10.3.0.4).

Variant 10.3.2.9. Let $\operatorname{\mathcal{C}}$ be a category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If $f$ is a quotient morphism (in the sense of Definition 10.3.2.1), then it is an epimorphism.

Proof. Suppose that we are given a pair of morphisms $e_0, e_1: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ satisfying $e_0 \circ f = e_1 \circ f$; we wish to show that $e_0 = e_1$. By virtue of our assumption that $f$ is a quotient morphism, it will suffice to show that $e_0 \circ h = e_1 \circ h$ for every morphism $h: C \rightarrow Y$ which belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ generated by $f$. In this case, we can write $h = f \circ g$ for some morphism $g: C \rightarrow X$; the desired result then follows from the calculation

\[ e_0 \circ h = e_0 \circ (f \circ g) = (e_0 \circ f) \circ g = (e_1 \circ f) \circ g = e_1 \circ (f \circ g) = e_1 \circ h. \]
$\square$

Warning 10.3.2.10. Let $f: X \rightarrow Y$ be a quotient morphism in an $\infty $-category $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ is not (the nerve of) an ordinary category, then $f$ need not be an epimorphism. For example, let $(X,x)$ be a pointed Kan complex, and let $\iota : \{ x\} \rightarrow X$ denote the inclusion map, which we regard as a morphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces. Then:

  • The morphism $\iota $ is an epimorphism (in the $\infty $-category $\operatorname{\mathcal{S}}$) if and only if $X$ is contractible. To see this, we observe that the identity map ${\operatorname{id}}_{X}$ and the constant map $c: X \rightarrow \{ x\} \xrightarrow {\iota } X$ become homotopic after precomposition with $\iota $; if $\iota $ is an epimorphism, it follows that ${\operatorname{id}}_{X}$ is homotopic to $c$.

  • The morphism $\iota $ is a quotient morphism if and only if $X$ is connected (see Proposition 10.3.4.17).

Proposition 10.3.2.11 (Homotopy Invariance). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Then $f$ is a quotient morphism if and only if $F(f)$ is a quotient morphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

Proof. Let $\operatorname{\mathcal{C}}^{0}_{ / Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve generated by $f$, and let $\operatorname{\mathcal{D}}^{0}_{ / F(Y) } \subseteq \operatorname{\mathcal{D}}^{0}_{ / F(Y) }$ be the sieve generated by $F(f)$. Since $F$ is fully faithful, $\operatorname{\mathcal{C}}^{0}_{/Y}$ is the inverse image of $\operatorname{\mathcal{D}}^{0}_{ / F(Y) }$ under the functor $F_{/Y}: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / F(Y) }$ induced by $F$. Corollary 4.6.4.19 guarantees that $F_{/Y}$ is an equivalence of $\infty $-categories, and therefore induces an equivalence $F_{/Y}^{0}: \operatorname{\mathcal{C}}_{/Y}^{0} \rightarrow \operatorname{\mathcal{D}}^{0}_{/F(Y)}$ (Corollary 4.5.2.29). In particular, $F_{/Y}^{0}$ is right cofinal (Corollary 7.2.1.13). Applying Corollary 7.2.2.3, we deduce that $F(f)$ is a quotient morphism if and only if the composite functor

\[ ( \operatorname{\mathcal{C}}_{ / Y}^{0} )^{\triangleright } \hookrightarrow ( \operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]

is a colimit diagram in $\operatorname{\mathcal{D}}$. By virtue of Proposition 7.1.4.10, this is equivalent to the requirement that $f$ is a quotient morphism in $\operatorname{\mathcal{C}}$. $\square$

Corollary 10.3.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0$ and $f_1$ be morphisms of $\operatorname{\mathcal{C}}$ which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$). Then $f_0$ is a quotient morphism if and only if $f_1$ is a quotient morphism.

Proof. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms. By virtue of Corollary 4.4.5.10, the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ are equivalences of $\infty $-categories. Our assumption that $f_0$ is isomorphic to $f_1$ guarantees that there exists a morphism $\widetilde{f}$ of $\operatorname{Isom}(\operatorname{\mathcal{C}})$ satisfying $\operatorname{ev}_0( \widetilde{f} ) = f_0$ and $\operatorname{ev}_1( \widetilde{f} ) = f_1$. Using Proposition 10.3.2.11, we see that the condition that $f_0$ is a quotient morphism in $\operatorname{\mathcal{C}}$ is equivalent to the condition that $\widetilde{f}$ is a quotient morphism in $\operatorname{Isom}(\operatorname{\mathcal{C}})$, which is also equivalent to the condition that $f_1$ is a quotient morphism in $\operatorname{\mathcal{C}}$. $\square$

Example 10.3.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of morphisms $f_0, f_1: X \rightarrow Y$ which are homotopic. Then $f_0$ is a quotient morphism if and only if $f_1$ is a quotient morphism. This is a special case of Corollary 10.3.2.12, but can also be deduced immediately from the definition (since $f_0$ and $f_1$ generate the same sieve on $Y$).

Proposition 10.3.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{/q}$ having image $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. If $f$ is a quotient morphism in $\operatorname{\mathcal{C}}$, then $\widetilde{f}$ is a quotient morphism in $\operatorname{\mathcal{C}}_{/q}$.

Proof. Set $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}_{/q}$, so that we have a commutative diagram of forgetful functors

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}}_{ / \widetilde{Y} } \ar [d]^{ \widetilde{U} } \ar [r]^-{V'} & \operatorname{\mathcal{C}}_{/Y} \ar [d]^{U} \\ \widetilde{\operatorname{\mathcal{C}}} \ar [r]^-{V} & \operatorname{\mathcal{C}}. } \]

Let $\operatorname{\mathcal{C}}^{0}_{ / Y } \subseteq \operatorname{\mathcal{C}}_{/Y}$ denote the sieve generated by $f$, so that $\widetilde{\operatorname{\mathcal{C}}}^{0}_{ / \widetilde{Y} } = V'^{-1} \operatorname{\mathcal{C}}^{0}_{/Y}$ is the sieve generated by $\widetilde{f}$. Note that $V$ is a right fibration (Proposition 4.3.6.1), so that $V'$ is a trivial Kan fibration (Corollary 4.3.7.13). In particular, the induced map $\widetilde{\operatorname{\mathcal{C}}}^{0}_{/ \widetilde{Y} } \rightarrow \operatorname{\mathcal{C}}^{0}_{/Y}$ is a trivial Kan fibration, and therefore right cofinal (Corollary 7.2.1.13). Combining our assumption that $f$ is a quotient morphism with Corollary 7.2.2.3, we deduce that the composite functor

\[ ( \widetilde{\operatorname{\mathcal{C}}}_{ / \widetilde{Y}}^{0} )^{\triangleright } \hookrightarrow ( \widetilde{\operatorname{\mathcal{C}}}_{/\widetilde{Y}})^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{C}}} \xrightarrow {V} \operatorname{\mathcal{C}} \]

is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Since the functor $V$ is conservative and creates colimits (Proposition 7.1.4.20), we conclude that $\widetilde{f}$ is a quotient morphism. $\square$

We close this section by recording two negative results, highlighting that the collection of quotient morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$ has poor closure properties in general:

  • The collection of quotient morphisms need not be closed under composition (Exercise 10.3.2.15).

  • The collection of quotient morphisms need not be closed under the formation of pullbacks (Exercise 10.3.2.16).

Both of these defects can be remedied by working instead with the class of universal quotient morphisms, which we study in §10.3.4 (see Definition 10.3.4.1).

Exercise 10.3.2.15. Let $\operatorname{\mathcal{C}}$ be the (nerve of the) ordinary category depicted informally by the diagram

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{X} \ar@ <-.4ex>[d]_{ e_0 } \ar@ <.4ex>[d]^{e_1} \ar [dr] & \widetilde{Y} \ar@ <-.4ex>[d]_{ g_0 } \ar@ <.4ex>[d]^{g_1} \ar [dr] & \\ X \ar [r]^-{ f } & Y \ar [r]^-{h} & Z, } \]

so that $f \circ e_0 = f \circ e_1$ and $h \circ g_0 = h \circ g_1$. Show that $f$ and $h$ are quotient morphisms in $\operatorname{\mathcal{C}}$, but the composition $(h \circ f): X \rightarrow Z$ is not a quotient morphism.

Exercise 10.3.2.16. Let $\operatorname{\mathcal{C}}$ be the category of partially ordered sets (where morphisms are nondecreasing functions). Let $Q = \{ a, b, c, d \} $ be a set with four elements, endowed with the partial ordering indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ a \ar [r] & b & c \ar [l] \ar [r] & d. } \]

Let $f: Q \rightarrow [2] = \{ 0 < 1 < 2 \} $ be the nondecreasing function given by

\[ f(a) = 0 \quad \quad f(b) = 1 = f(c) \quad \quad f(d) = 2, \]

so that we have a pullback diagram of partially ordered sets

10.21
\begin{equation} \begin{gathered}\label{equation:poset-not-universal} \xymatrix@R =50pt@C=50pt{ \{ a, d \} \ar [r] \ar [d]^{f_0} & Q \ar [d]^{f} \\ \{ 0 < 2 \} \ar [r] & [2]. } \end{gathered} \end{equation}

Show that $f$ is a quotient morphism in (the nerve of) the category $\operatorname{\mathcal{C}}$, but that $f_0$ is not.

Variant 10.3.2.17. In the situation of Exercise 10.3.2.16, we can apply the nerve functor to (10.21) and obtain a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \{ a, d \} \ar [r] \ar [d]^{F_0} & \operatorname{N}_{\bullet }(Q) \ar [d]^{F} \\ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \ar [r] & \Delta ^2, } \]

which we can regard as a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. Show that $F$ is a quotient morphism in $\operatorname{\mathcal{QC}}$, but that $F_0$ is not (beware that this is not a formal consequence of Exercise 10.3.2.16: the construction $P \mapsto \operatorname{N}_{\bullet }(P)$ does not preserve quotient morphisms in general).