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7.6.5 Equalizers and Coequalizers

We now study (co)limits of a particularly simple shape.

Notation 7.6.5.1. Let $( \bullet \rightrightarrows \bullet )$ denote the simplicial set given by the pushout $\Delta ^1 \coprod _{ \operatorname{\partial \Delta }^1 } \Delta ^1$. For any $\infty $-category $\operatorname{\mathcal{C}}$, we will identify morphisms from $( \bullet \rightrightarrows \bullet )$ to $\operatorname{\mathcal{C}}$ with pairs $(f_0, f_1)$, where $f_0: Y \rightarrow X$ and $f_1: Y \rightarrow X$ are morphisms of $\operatorname{\mathcal{C}}$ having the same source and target.

Remark 7.6.5.2. The simplicial set $( \bullet \rightrightarrows \bullet )$ of Notation 7.6.5.1 is isomorphic to the nerve of its homotopy category $\operatorname{\mathcal{J}}$, which can be described concretely as follows:

  • The category $\operatorname{\mathcal{J}}$ has exactly two objects $Y$ and $X$.

  • There are exactly two non-identity morphisms in $\operatorname{\mathcal{J}}$, both of which have source $Y$ and target $X$.

Remark 7.6.5.3. There is a tautological epimorphism of simplicial sets

\begin{eqnarray*} ( \bullet \rightrightarrows \bullet ) & = & \Delta ^1 \coprod _{ \operatorname{\partial \Delta }^1 } \Delta ^1 \\ & \twoheadrightarrow & \Delta ^1 \coprod _{ \operatorname{\partial \Delta }^1 } \Delta ^0 \\ & = & \Delta ^1 / \operatorname{\partial \Delta }^1. \end{eqnarray*}

It follows from Example 6.3.4.4 that this epimorphism exhibits $\Delta ^1 / \operatorname{\partial \Delta }^1$ as a localization of $( \bullet \rightrightarrows \bullet )$. In particular, it is both left and right cofinal (Proposition 7.2.1.9).

Definition 7.6.5.4 (Equalizers and Coequalizers). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0, f_1: Y \rightarrow X$ be morphisms of $\operatorname{\mathcal{C}}$ having the same source and target, which we identify with functor $\sigma : ( \bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{C}}$. An equalizer of $f_0$ and $f_1$ is a limit of the diagram $\sigma $. A coequalizer of $f_0$ and $f_1$ is a colimit of the diagram $\sigma $. We say that the $\infty $-category $\operatorname{\mathcal{C}}$ admits equalizers if every pair of morphisms $f_0, f_1: Y \rightarrow X$ have an equalizer in $\operatorname{\mathcal{C}}$, and that $\operatorname{\mathcal{C}}$ admits coequalizers if every pair of morphisms $f_0, f_1: Y \rightarrow X$ have a coequalizer in $\operatorname{\mathcal{C}}$.

Notation 7.6.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0, f_1: Y \rightarrow X$ be morphisms of $\operatorname{\mathcal{C}}$ having the same source and target. If there exists an object $Z \in \operatorname{\mathcal{C}}$ which is an equalizer of $f_0$ and $f_1$, then $Z$ is uniquely determined up to isomorphism (Proposition 7.1.1.12). To emphasize this uniqueness, we denote the object $Z$ (if it exists) by $\operatorname{Eq}(f_0, f_1)$. Similarly, if there exists an object $W \in \operatorname{\mathcal{C}}$ which is a coequalizer of $f_0$ and $f_1$, then $W$ is uniquely determined up to isomorphism; to emphasize this, we denote $W$ by $\operatorname{Coeq}(f_0,f_1)$.

Remark 7.6.5.6 (Duality). The simplicial set $( \bullet \rightrightarrows \bullet )$ is canonically isomorphic to its opposite $( \bullet \rightrightarrows \bullet )^{\operatorname{op}}$. Consequently, if $f_0, f_1: Y \rightarrow X$ are morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$ which admit an equalizer $Z = \operatorname{Eq}(f_0, f_1)$, then $Z$ can be regarded as a coequalizer of $f_0$ and $f_1$ in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 7.6.5.7 (Symmetry). The simplicial set $( \bullet \rightrightarrows \bullet )$ has a unique nontrivial automorphism, which exchanges its nondegenerate edges. It follows that, if $f_0, f_1: Y \rightarrow X$ are a pair of morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$, then we can identify (co)equalizers of the pair $(f_0, f_1)$ with (co)equalizers of the pair $(f_1, f_0)$.

Example 7.6.5.8 (Fixed Points of Endomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. An endomorphism of $X$ is a morphism $f: X \rightarrow X$ from the object $X$ to itself. Note that the pair $(X,f)$ can be identified with a morphism of simplicial sets $\sigma : (\Delta ^1 / \operatorname{\partial \Delta }^1) \rightarrow \operatorname{\mathcal{C}}$. It follows from Remark 7.6.5.3 (together with Corollary 7.2.2.11) that an object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $\sigma $ if and only if it is an equalizer of the pair of morphisms $f, \operatorname{id}_{X}: X \rightarrow X$. Similarly, an object of $\operatorname{\mathcal{C}}$ is a colimit of $\sigma $ if and only if it is a coequalizer of the pair $(f, \operatorname{id}_ X)$.

Variant 7.6.5.9. Let $\operatorname{\mathbf{Z}}_{\geq 0}$ denote the collection of nonnegative integers, which we regard as a commutative monoid under addition, and let $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ denote the classifying simplicial set of Example 1.2.4.3. The simplicial set $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ is an $\infty $-category which contains a (unique) object $X$, and the generator $1 \in \operatorname{\mathbf{Z}}_{\geq 0}$ determines an endomorphism $e: X \rightarrow X$. We can regard $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ as freely generated by the endomorphism $e$: more precisely, the pair $(X,e)$ determines a morphism of simplicial sets $\sigma : \Delta ^1 / \operatorname{\partial \Delta }^1 \hookrightarrow B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ which is inner anodyne (see Example 1.4.7.10), and therefore induces a trivial Kan fibration $\operatorname{Fun}( B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$ for every $\infty $-category $\operatorname{\mathcal{C}}$. In particular, the morphism $\sigma $ is both left and right cofinal (Proposition 7.2.1.3).

If $F: B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathcal{C}}$ is a functor of $\infty $-categories, then Corollary 7.2.2.11 guarantees that an object of $\operatorname{\mathcal{C}}$ is a limit of the functor $F$ if and only if it is a limit of the diagram $F \circ \sigma $: that is, if and only if it is an equalizer of the pair of morphisms $F(e), \operatorname{id}_{F(X)}: F(X) \rightarrow F(X)$ (see Example 7.6.5.8). Similarly, an object of $\operatorname{\mathcal{C}}$ is a colimit of the functor $F$ if and only if it is a coequalizer of the pair $( F(e), \operatorname{id}_{ F(X)} )$.

Definition 7.6.5.10 (Equalizer and Coequalizer Diagrams). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An equalizer diagram in $\operatorname{\mathcal{C}}$ is a limit diagram $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. A coequalizer diagram is a colimit diagram $( \bullet \rightrightarrows \bullet )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Warning 7.6.5.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given an equalizer diagram

7.63
\begin{equation} \begin{gathered}\label{equation:equalizer-not-in-homotopy} \xymatrix { Z \ar [r]^-{g} & Y \ar@ <.4ex>[r]^-{f_0} \ar@ <-.4ex>[r]_-{f_1} & X } \end{gathered} \end{equation}

in $\operatorname{\mathcal{C}}$. Then the image of (7.63) in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ need not be an equalizer diagram. In other words, the forgetful functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}})$ does not preserve equalizer diagrams in general.

Example 7.6.5.12. Let $X$ be a Kan complex containing vertices $x$ and $y$. Then there exists an equalizer diagram

7.64
\begin{equation} \begin{gathered}\label{equation:bad-equalizer} \xymatrix { \{ x\} \times ^{\mathrm{h}}_{X} \{ y\} \ar [r]^-{f} & \Delta ^0 \ar@ <.4ex>[r]^-{x} \ar@ <-.4ex>[r]_-{y} & X } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{\mathcal{S}}$ (for a more general statement, see Corollary 7.6.5.21). However, unless the homotopy fiber product $\{ x\} \times ^{\mathrm{h}}_{X} \{ y\} $ is either empty or contractible, the image of (7.64) in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ is not an equalizer diagram (since the homotopy class $[f]$ is not a monomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$).

Exercise 7.6.5.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given an equalizer diagram

7.65
\begin{equation} \begin{gathered}\label{equation:half-equalizer-always} \xymatrix { Z \ar [r]^-{g} & Y \ar@ <.4ex>[r]^-{f_0} \ar@ <-.4ex>[r]_-{f_1} & X } \end{gathered} \end{equation}

in $\operatorname{\mathcal{C}}$. Show that, for every object $C \in \operatorname{\mathcal{C}}$, the map of sets

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(C,Z) \xrightarrow { [g] \circ } \operatorname{Eq}( \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(C,Y) \rightrightarrows \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X) ) \]

is surjective (though it is generally not injective).

We now give some examples of (co)equalizer diagrams.

Proposition 7.6.5.14. Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories satisfying $F_0 \circ G = F_1 \circ G$. The following conditions are equivalent:

$(1)$

The resulting diagram of $\infty $-categories $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ is an equalizer diagram.

$(2)$

The commutative diagram

7.66
\begin{equation} \begin{gathered}\label{equation:homotopy-equalizer-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{G} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^-{ (F_0, F_1) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id}, \operatorname{id}) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

is a categorical pullback square (Definition 4.5.2.7).

Proof. Let us identify the pair $(F_0, F_1)$ with a functor of ordinary categories $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{QCat}$, where $\operatorname{\mathcal{J}}$ is the category described in Remark 7.6.5.2. The functor $G$ then induces a map $\operatorname{\mathcal{E}}\rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$, which can be identified with the map

\[ \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}} \]

determined by the diagram (7.66). Proposition 7.6.5.14 now follows from the criterion of Corollary 7.5.5.8. $\square$

Corollary 7.6.5.15. Let $f_0, f_1: Y \rightarrow X$ be morphisms of Kan complexes and let $g: Z \rightarrow Y$ be a morphism of Kan complexes satisfying $f_0 \circ g = f_1 \circ g$. The following conditions are equivalent:

$(1)$

The resulting diagram of $\infty $-categories $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is an equalizer diagram.

$(2)$

The commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ Z \ar [r]^-{g} \ar [d] & Y \ar [d]^-{ (f_0, f_1) } \\ X \ar [r]^-{ (\operatorname{id}, \operatorname{id}) } & X \times X } \]

is a homotopy pullback square.

Corollary 7.6.5.16. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and suppose we are given morphisms

\[ \xymatrix@R =50pt@C=50pt{ Z \ar [r]^-{g} & Y \ar@ <.4ex>[r]^-{f_0} \ar@ <-.4ex>[r]_-{f_1} & X } \]

in $\operatorname{\mathcal{C}}$ satisfying $f_0 \circ g = f_1 \circ g$. The following conditions are equivalent:

$(1)$

The induced diagram $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an equalizer diagram in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, in the sense of Definition 7.6.5.10.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Z)_{\bullet } \ar [r] \ar [d]^-{g} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y )_{\bullet } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } \ar [r]^-{(f_0, f_1) } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } } \]

is a homotopy pullback square.

Let $f_0, f_1: Y \rightarrow X$ be morphisms of Kan complexes. By virtue of Corollary 7.4.5.6, the morphisms $f_0$ and $f_1$ have an equalizer in the $\infty $-category $\operatorname{\mathcal{S}}$. Beware that this equalizer generally cannot be obtained from Corollary 7.6.5.15. For example, if $f_0$ and $f_1$ have disjoint images, then the existence of a morphism $g: Z \rightarrow Y$ satisfying $f_0 \circ g = f_1 \circ g$ guarantees that the simplicial set $Z$ is empty. In such cases, to extend the pair $(f_0, f_1)$ to an equalizer diagram in $\operatorname{\mathcal{S}}$, we are forced to consider homotopy coherent diagrams which do not strictly commute.

Remark 7.6.5.17. Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, which we identify with a diagram $\sigma : ( \bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{QC}}$. Unwinding the definitions, we see that extensions of $\sigma $ to a diagram $\overline{\sigma }: ( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ can be identified with the following data:

  • An $\infty $-category $\operatorname{\mathcal{E}}$ equipped with functors $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $H: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

  • Isomorphisms $\alpha _0: F_0 \circ G \xrightarrow {\sim } H$ and $\alpha _1: F_1 \circ G \xrightarrow {\sim } H$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$.

In this case, we can identify the quadruple $(G, H, \alpha _0, \alpha _1)$ with a single functor of $\infty $-categories

\[ U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]

Proposition 7.6.5.18. Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, which we identify with a diagram $\sigma : ( \bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{QC}}$. Suppose we are given an extension $\overline{\sigma }: ( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ of $\sigma $, corresponding to a functor of $\infty $-categories

\[ T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]

Then $\overline{\sigma }$ is an equalizer diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if $T$ is an equivalence of $\infty $-categories.

Proof. We proceed as in the proof of Proposition 7.6.4.8, with minor modifications. Let $\operatorname{\mathcal{A}}$ denote the simplicial path category of $( \bullet \rightrightarrows \bullet )^{\triangleleft }$, so that we can identify $\overline{\sigma }$ with a simplicial functor $\mathscr {F}: \operatorname{\mathcal{A}}\rightarrow \operatorname{QCat}$. Using Corollary 5.3.7.5, we can factor the functor $(F_0, F_1): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}$ as a composition

\[ \operatorname{\mathcal{D}}\xrightarrow { U } \operatorname{\mathcal{D}}' \xrightarrow { (F'_0, F'_1) } \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}, \]

where $U$ is an equivalence of $\infty $-categories and $(F'_0, F'_1)$ is an isofibration. The pair $(F'_0, F'_1)$ can be identified with a morphism of simplicial sets $\sigma ': (\bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{QC}}$. Applying Remark 7.6.5.17, we can extend $\sigma '$ to a diagram $\overline{\sigma }': (\bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$, carrying the cone point to the $\infty $-category $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{D}}' \times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$. The diagram $\overline{\sigma }'$ corresponds to a simplicial functor $\mathscr {F}': \operatorname{\mathcal{A}}\rightarrow \operatorname{QCat}$. The morphisms $T$ and $U$ determine a natural transformation of simplicial functors $\mathscr {F} \rightarrow \mathscr {F}'$, hence also a morphism $\overline{\sigma } \rightarrow \overline{\sigma }'$ in the $\infty $-category $\operatorname{Fun}( ( \bullet \rightrightarrows \bullet )^{\triangleleft }, \operatorname{\mathcal{QC}})$. By virtue of Corollary 4.5.2.18, this natural transformation is an isomorphism of diagrams if and only if the functor $T$ is an equivalence of $\infty $-categories. Consequently, Proposition 7.6.5.18 is equivalent to the assertion that $\overline{\sigma }'$ is an equalizer diagram in $\operatorname{\mathcal{QC}}$ (Proposition 7.1.2.13).

Invoking Remark 7.6.5.17 again, we obtain another diagram $\overline{\sigma }''$ extending $\sigma '$, which carries the cone point of $(\bullet \rightrightarrows \bullet )^{\triangleleft }$ to the equalizer $\operatorname{Eq}(F'_0, F'_1) = \operatorname{\mathcal{D}}' \times _{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ (formed in the category of simplicial sets). The diagram $\overline{\sigma }''$ corresponds to another simplicial functor $\mathscr {F}'': \operatorname{\mathcal{A}}\rightarrow \operatorname{QCat}$. Note that there is a natural inclusion map $\mathscr {F}'' \hookrightarrow \mathscr {F}'$, which carries the cone point to the inclusion

\[ \iota : \operatorname{\mathcal{D}}' \times _{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}' \times _{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) }^{\mathrm{h}} \operatorname{\mathcal{C}}. \]

Since $(F'_0, F'_1)$ is an isofibration, the functor $\iota $ is an equivalence of $\infty $-categories (Corollary 4.5.2.22). It follows that the inclusion $\mathscr {F}'' \hookrightarrow \mathscr {F}'$ induces an isomorphism $\overline{\sigma }'' \rightarrow \overline{\sigma }'$ in the $\infty $-category $\operatorname{Fun}( ( \bullet \rightrightarrows \bullet )^{\triangleleft }, \operatorname{\mathcal{QC}})$. By virtue of Proposition 7.1.2.13, we are reduced to showing that $\overline{\sigma }''$ is an equalizer diagram in $\operatorname{\mathcal{QC}}$. This follows from the criterion of Proposition 7.6.5.14 (since $\iota $ is an equivalence of $\infty $-categories). $\square$

It follows from Proposition 7.6.5.18 that, if $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors of $\infty $-categories, then the homotopy fiber product $\operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is an equalizer of $F_0$ and $F_1$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ (this can also be viewed as a special case of Proposition 7.5.2.4). However, it is possible to be more efficient.

Construction 7.6.5.19 (The Homotopy Equalizer). Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and form a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{hEq}(F_0, F_1) \ar [r] \ar [d]^-{G} & \operatorname{Isom}(\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^-{ (F_0, F_1) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}. } \]

Note that the right vertical map is an isofibration (Corollary 4.4.5.5), so the left vertical map is also an isofibration; in particular, $\operatorname{hEq}(F_0, F_1)$ is an $\infty $-category. We will refer to $\operatorname{hEq}( F_0, F_1 )$ as the homotopy equalizer of the functors $F_0$ and $F_1$. By construction, objects of $\operatorname{hEq}(F_0, F_1)$ can be identified with pairs $(X, u)$, where $X$ is an object of $\operatorname{\mathcal{D}}$ and $u: F_0(X) \rightarrow F_1(X)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

Set $H = G \circ F_1$, so that the construction $(X,u) \mapsto u$ determines an isomorphism $\alpha _0: G \circ F_0 \xrightarrow {\sim } H$ in the $\infty $-category $\operatorname{Fun}( \operatorname{Eq}(F_0, F_1), \operatorname{\mathcal{C}})$. Taking $\alpha _1$ to be the identity morphism $\operatorname{id}: G \circ F_1 \xrightarrow {\sim } H$, we see that the quadruple $(G,H, \alpha _0, \alpha _1 )$ determines a diagram $\overline{\sigma }: ( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$, carrying the cone point to the homotopy equalizer $\operatorname{hEq}(F_0, F_1)$ (see Remark 7.6.5.17).

Corollary 7.6.5.20. Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Then the morphism $\overline{\sigma }: ( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ of Construction 7.6.5.19 is an equalizer diagram. In particular, the homotopy equalizer $\operatorname{hEq}( F_0, F_1)$ is an equalizer of $F_0$ and $F_1$ in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proof. The diagram $\overline{\sigma }$ can be identified with a functor $U: \operatorname{hEq}( F_0, F_1 ) \rightarrow \operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$. By virtue of Proposition 7.6.5.18, it will suffice to show that $U$ is an equivalence of $\infty $-categories. Unwinding the definitions, we see that $U$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{hEq}( F_0, F_1) \ar [r]^-{U} \ar [d] & \operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{(\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}\ar [r] \ar [d] & \operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{D}}\times \operatorname{\mathcal{C}}, } \]

where the homotopy fiber product on the upper right is formed using the functor $F_0$, and the homotopy fiber product on the lower middle is formed using the functor $F_1$. Each of the squares in this diagram is a pullback, and the right vertical map is an isofibration (Remark 4.5.2.2). It follows that the left side of the diagram is a categorical pullback square (Corollary 4.5.2.21). Since the functor on the lower left is an equivalence of $\infty $-categories (Proposition 5.3.7.4), it follows that $U$ is an equivalence of $\infty $-categories. $\square$

Corollary 7.6.5.21. Let $f_0, f_1: Y \rightarrow X$ be morphisms of Kan complexes. Then the homotopy equalizer $\operatorname{hEq}(f_0, f_1)$ is a Kan complex, which is an equalizer of $f_0$ and $f_1$ in the $\infty $-category $\operatorname{\mathcal{S}}$.

Corollaries 7.6.5.20 and 7.6.5.21 illustrate a general phenomenon: if $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits pairwise products, then equalizers in $\operatorname{\mathcal{C}}$ can be viewed as a special kind of fiber product.

Proposition 7.6.5.22 (Rewriting Equalizers as Pullbacks). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f_0, f_1: Y \rightarrow X$ be morphisms of $\operatorname{\mathcal{C}}$. Let $X \times X$ be a product of $X$ with itself in the $\infty $-category $\operatorname{\mathcal{C}}$, so that $f_0$ and $f_1$ determine a morphism $(f_0, f_1): Y \rightarrow X \times X$, and let $\delta _{X}: X \rightarrow X \times X$ be the diagonal map. Then an object of $\operatorname{\mathcal{C}}$ is an equalizer of $f_0$ and $f_1$ if and only if it is a fiber product of $Y$ with $X$ over $X \times X$.

Proof. Let $\operatorname{\mathcal{K}}$ denote the simplicial set given by the product $( \bullet \rightrightarrows \bullet )^{\triangleleft } \times \Delta ^1$. Then $\operatorname{\mathcal{K}}$ is an $\infty $-category, which we depict informally by the diagram

\[ \xymatrix@R =50pt@C=50pt{ z \ar [r] \ar [d] & y \ar@ <.4ex>[r] \ar@ <-.4ex>[r] \ar [d] & x \ar [d] \\ z' \ar [r] & y' \ar@ <.4ex>[r] \ar@ <-.4ex>[r] & x'. } \]

We now proceed in several steps.

  • Let $\operatorname{\mathcal{K}}_{0}$ denote the full subcategory of $\operatorname{\mathcal{K}}$ spanned by the objects $x$ and $y$. Then $\operatorname{\mathcal{K}}_0$ is isomorphic to the simplicial set $( \bullet \rightrightarrows \bullet )$. In particular, the pair of morphisms $f_0, f_1: Y \rightarrow X$ can be identified with a functor $\sigma _0: \operatorname{\mathcal{K}}_0 \rightarrow \operatorname{\mathcal{C}}$, satisfying $\sigma _0( x ) = X$ and $\sigma _0( y ) = Y$. By definition, an object of $\operatorname{\mathcal{C}}$ is an equalizer of the pair $(f_0, f_1)$ if and only if it is a limit of the diagram $\sigma _0$.

  • Let $\operatorname{\mathcal{K}}_1$ denote the full subcategory of $\operatorname{\mathcal{K}}$ spanned by the objects $x$, $x'$, and $y$. Note that the identity map $\operatorname{id}_{ \operatorname{\mathcal{K}}_0 }$ extends uniquely to a retraction $r: \operatorname{\mathcal{K}}_1 \rightarrow \operatorname{\mathcal{K}}_0$, carrying the object $x' \in \operatorname{\mathcal{K}}_1$ to $x \in \operatorname{\mathcal{K}}_0$. Let $\sigma _1: \operatorname{\mathcal{K}}_1 \rightarrow \operatorname{\mathcal{C}}$ be the composition $\sigma _0 \circ r$. Note that that the inclusion map $\operatorname{\mathcal{K}}_0 \hookrightarrow \operatorname{\mathcal{K}}_1$ admits a right adjoint (given by the retraction $r$), and is therefore left cofinal (Corollary 7.2.3.6). It follows that an object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $\sigma _0$ if and only if it is a limit of the diagram $\sigma _1$ (Corollary 7.2.2.11).

  • Choose a pair of morphisms $\pi _0, \pi _1: X \times X \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ which exhibit $X \times X$ as a product of $X$ with itself. The morphism $(f_0, f_1): Y \rightarrow X$ is characterized (up to homotopy) by the requirement that there exist $2$-simplices $\sigma _0$ and $\sigma _1$ of $\operatorname{\mathcal{C}}$, where $\sigma _ i$ exhibits $f_ i$ as a composition of $\pi _ i$ with $(f_0, f_1)$. Let $\operatorname{\mathcal{K}}_2$ denote the full subcategory of $\operatorname{\mathcal{K}}$ spanned by the objects $x$, $x'$, $y$, and $y'$. Then the pair $(\sigma _0, \sigma _1)$ determines an extension of $\sigma _1$ to a functor $\sigma _2: \operatorname{\mathcal{K}}_2 \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma _2(y') = X \times X$.

  • The diagonal morphism $\delta _{X}: X \rightarrow X \times X$ is characterized (up to homotopy) by the requirement that there exist $2$-simplices $\tau _0$ and $\tau _ i$ of $\operatorname{\mathcal{C}}$, where $\tau _ i$ exhibits $\operatorname{id}_{X}$ as a composition of $\pi _ i$ with $\delta _{X}$. Let $\operatorname{\mathcal{K}}_3$ denote the full subcategory of $\operatorname{\mathcal{K}}$ spanned by the objects $x$, $x'$, $y$, and $y'$, and $z'$. Then the pair $(\tau _0, \tau _1)$ determines an extension of $\sigma _2$ to a functor $\sigma _3: \operatorname{\mathcal{K}}_3 \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma _3(z') = X$. The diagram $\sigma _3$ can be represented informally by the diagram

    \[ \xymatrix@R =50pt@C=50pt{ \bullet \ar@ {-->}[r] \ar@ {-->}[d] & Y \ar@ <.4ex>[r]^{f_0} \ar@ <-.4ex>[r]_{f_1} \ar [d]^-{ (f_0, f_1) } & X \ar [d]^-{\operatorname{id}_ X} \\ X \ar [r]^-{\delta _ X} & X \times X \ar@ <.4ex>[r]^{\pi _0} \ar@ <-.4ex>[r]_{\pi _1} & X. } \]

    Note that $\sigma _3$ is right Kan extended from the full subcategory $\operatorname{\mathcal{K}}_1 \subseteq \operatorname{\mathcal{K}}_3$. Consequently, an object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $\sigma _1$ if and only if it is a limit of the diagram $\sigma _3$ (Remark 7.3.7.15).

  • Let $\operatorname{\mathcal{K}}_{4}$ denote the full subcategory of $\operatorname{\mathcal{K}}$ spanned by the objects $x'$, $y$, $y'$, and $z'$. Note that the functor $\sigma _3$ is right Kan extended from $\operatorname{\mathcal{K}}_4$. It follows that an object of $\operatorname{\mathcal{C}}$ is a limit of the functor $\sigma _3$ if and only if it is a limit of the functor $\sigma _4 = \sigma _3 |_{\operatorname{\mathcal{K}}_4}$.

  • Let $\operatorname{\mathcal{K}}_5$ denote the full subcategory of $\operatorname{\mathcal{K}}$ spanned by the objects $y$, $y'$, and $z'$. Using the criterion of Theorem 7.2.3.1, we see that the inclusion $\operatorname{\mathcal{K}}_{5} \hookrightarrow \operatorname{\mathcal{K}}_4$ is left cofinal. It follows that an object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $\sigma _4$ if and only if it is a limit of the diagram $\sigma _5 = \sigma _4 |_{ \operatorname{\mathcal{K}}_5}$ (Corollary 7.2.2.11).

Combining these steps, we deduce that an object of $\operatorname{\mathcal{C}}$ is an equalizer of $f_0$ and $f_1$ if and only if it is a limit of the diagram $\sigma _5$: that is, if and only if it is a fiber product of $Y$ with $X$ over $X \times X$ (along the morphisms $(f_0, f_1)$ and $\delta _ X$). $\square$

In an $\infty $-category which admits finite products, we can use a similar argument to describe pullbacks in terms of equalizers.

Proposition 7.6.5.23 (Rewriting Pullbacks as Equalizers). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be morphisms of $\operatorname{\mathcal{C}}$. Suppose that $X_0$ and $X_1$ admit a product $X_0 \times X_1$, and let $\pi _0: X_0 \times X_1 \rightarrow X_0$ and $\pi _1: X_0 \times X_1 \rightarrow X_1$ denote the projection maps. For $i \in \{ 0,1\} $, let $g_ i: X_0 \times X_1 \rightarrow X$ denote a composition of $\pi _ i$ with $f_ i$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Then an object of $\operatorname{\mathcal{C}}$ is a pullback of $X_0$ with $X_1$ over $X$ if and only if it is an equalizer of the pair of morphisms $(g_0, g_1)$.

Proof. Let $\operatorname{\mathcal{K}}$ denote the category which is freely generated by a non-commutative square, as indicated in the diagram

\[ \xymatrix@R =70pt@C=70pt{ Y_{01} \ar [r] \ar [d] \ar@ <.8ex>[dr] \ar@ <-.8ex>[dr] & Y_0 \ar [d] \\ Y_1 \ar [r] & Y. } \]

Note that the upper right and lower left regions of this diagram determine monomorphisms $\tau _0, \tau _1: \Delta ^2 \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$. The images of $\tau _0$ and $\tau _1$ are simplicial subsets of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$, whose union is $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ and whose intersection is the discrete simplicial set $\{ Y_{01}, Y \} $. It follows that $\tau _0$ and $\tau _1$ induce an isomorphism of simplicial sets $(\tau _0, \tau _1): \Delta ^2 \coprod _{ \{ 0,2\} } \Delta ^2 \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$.

For $i \in \{ 0,1\} $, let $\sigma _ i$ be a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $g_ i$ as a composition of $\pi _ i$ with $f_ i$ (in the sense of Definition 1.3.4.1). Then there is a unique morphism of simplicial sets $q: \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $q \circ \tau _ i = \sigma _ i$, which we indicate as a diagram

\[ \xymatrix@R =70pt@C=70pt{ X_0 \times X_1 \ar [r]^-{\pi _0} \ar [d]_-{\pi _1} \ar@ <.8ex>[dr]^{g_0} \ar@ <-.8ex>[dr]_{g_1} & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. } \]

Let $\operatorname{\mathcal{K}}_{+} \subseteq \operatorname{\mathcal{K}}$ denote the full subcategory spanned by the objects $Y_{01}$ and $Y$. Then the nerve $\operatorname{N}_{\bullet }( \operatorname{\mathcal{K}}_{+} )$ can be identified with the simplicial set $( \bullet \rightrightarrows \bullet )$ of Notation 7.6.5.1, and the restriction $q_{+} = q|_{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{K}}_{+} ) }$ corresponds to the pair of morphisms $g_0, g_1: X_0 \times X_1 \rightarrow X$. Note that the full subcategory $\operatorname{N}_{\bullet }( \operatorname{\mathcal{K}}_{+} ) \subset \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ is coreflective, so the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}_{+}) \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ is left cofinal (Corollary 7.2.3.6). It follows that an object of $\operatorname{\mathcal{C}}$ is an equalizer of $g_0$ and $g_1$ if and only if it is a limit of the diagram $q$ (Corollary 7.2.2.11).

To complete the proof, it will suffice to show that an object of $\operatorname{\mathcal{C}}$ is a limit of $q$ if and only if it is a fiber product of $X_0$ with $X_1$ over $X$. Let $\operatorname{\mathcal{K}}_{-} \subseteq \operatorname{\mathcal{K}}$ denote the full subcategory spanned by the objects $Y_0$, $Y_1$, and $Y$. By virtue of Corollaries 7.3.7.2 and 7.3.7.14, it will suffice to show that the functor $q$ is right Kan extended from $\operatorname{N}_{\bullet }( \operatorname{\mathcal{K}}_{-} )$. E quivalently, we wish to show that the natural map

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{K}}_{-} \times _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}_{ Y_{01} / } )^{\triangleleft } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow {q} \operatorname{\mathcal{C}} \]

is a limit diagram in $\operatorname{\mathcal{C}}$. Unwinding the definitions, we see that $\operatorname{\mathcal{K}}_{-} \times _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}_{ Y_{01}/ }$ can be written as a disjoint union of subcategories having initial objects $Y_0$ and $Y_1$, respectively. In particular, the inclusion map

\[ \{ Y_0, Y_1 \} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{K}}_{-} \times _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}_{ Y_{01} / } ) \]

is left cofinal. The desired result now follows from Corollary 7.2.2.3, together with our assumption that the maps $\pi _0$ and $\pi _1$ exhibit $X_0 \times X_1$ as a product of $X_0$ with $X_1$. $\square$

Exercise 7.6.5.24. In the situation of Proposition 7.6.5.23, suppose that $X_0$ and $X_1$ admit a fiber product over $X$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves the product of $X_0$ and $X_1$ (that is, $F( \pi _0 )$ and $F( \pi _1 )$ exhibit $F( X_0 \times X_1 )$ as a product of $F(X_0)$ with $F(X_1)$ in the $\infty $-category $\operatorname{\mathcal{D}}$). Show that $F$ preserves the fiber product of $X_0$ with $X_1$ over $X$ if and only if it preserves the equalizer of the morphisms $g_0$ and $g_1$.

Corollary 7.6.5.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits finite limits if and only if it admits finite products and equalizers. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preservse finite limits if and only if it preseves finite products and equalizers.