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5.1.5 The Comma Construction

Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories. To every pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, one can associate the comma category $(F \downarrow G)$, whose objects are triples $(C, D, \eta )$ where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $\eta : F(C) \rightarrow G(D)$ is a morphism in the category $\operatorname{\mathcal{E}}$ (Notation 2.1.4.19). This construction has an obvious analogue in the setting of $\infty $-categories. We now generalize this construction to the setting of $\infty $-categories.

Definition 5.1.5.1 (The Comma Construction). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $(F \downarrow G)$ denote the simplicial set given by the iterated fiber product

\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}. \]

We will refer to $(F \downarrow G)$ as the comma construction on $F$ and $G$.

In the special case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{E}}$ and $F = \operatorname{id}_{\operatorname{\mathcal{C}}}$ is the identity map, we will denote the comma construction $(F \downarrow G)$ by $(\operatorname{\mathcal{C}}\downarrow G)$; it can be described more simply as the fiber product $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}}$. Similarly, in the special case where $\operatorname{\mathcal{D}}= \operatorname{\mathcal{E}}$ and $G = \operatorname{id}_{\operatorname{\mathcal{D}}}$ is the identity map, we denote the comma construction $(F \downarrow G)$ by $(F \downarrow \operatorname{\mathcal{D}})$; it can be described more simply as the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{D}}) } \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$.

As our notation suggests, we will be primarily interested in the special case of Definition 5.1.5.1 where the simplicial sets $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories.

Proposition 5.1.5.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category, and suppose we are given morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then the projection map $\theta : (F \downarrow G) \rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of simplicial sets.

Proof. By construction, we have a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ (F \downarrow G) \ar [r] \ar [d]^{\theta } & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \ar [d]^{\theta _0} \\ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}). } \]

Since $\operatorname{\mathcal{E}}$ is an $\infty $-category, the restriction map $\theta _0$ is an isofibration of $\infty $-categories (Corollary 4.4.5.3). Invoking Remark 4.5.7.7, we conclude that $\theta $ is an isofibration of simplicial sets. $\square$

Corollary 5.1.5.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Then the comma construction $(F \downarrow G)$ is also an $\infty $-category.

Proof. By virtue of Proposition 5.1.5.2, the projection map $(F \downarrow G) \rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration. Since $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an $\infty $-category, it follows that $(F \downarrow G)$ is also an $\infty $-category (Remark 4.5.7.3). $\square$

Example 5.1.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between ordinary categories, and let $(F \downarrow G)$ denote the comma category of Notation 2.1.4.19. Since the nerve construction is compatible with the formation of inverse limits and functor categories, we have a canonical isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }( F \downarrow G) \simeq ( \operatorname{N}_{\bullet }(F) \downarrow \operatorname{N}_{\bullet }(G) ). \]

Consequently, Definition 5.1.5.1 can be viewed as a generalization of the classical comma construction in category theory.

Example 5.1.5.5. Let $X$, $Y$, and $Z$ be topological spaces, and suppose we are given a pair of continuous functions $f: X \rightarrow Z$ and $g: Y \rightarrow Z$. We let $(f \downarrow g)$ denote the set of all triples $(x,y,\eta )$ where $x$ is a point of $X$, $y$ is a point of $Y$, and $\eta : [0,1] \rightarrow Z$ is a continuous function satisfying $\eta (0) = f(x)$ and $\eta (1) = g(y)$. The set $(f \downarrow g)$ carries a natural topology, which is defined by viewing it as a subspace of the product $X \times Y \times \operatorname{Hom}_{\operatorname{Top}}( [0,1], Z)$ (where we endow the path space $\operatorname{Hom}_{\operatorname{Top}}([0,1],Z)$ with the compact-open topology). We then have a canonical isomorphism of simplicial sets

\[ \operatorname{Sing}_{\bullet }( f \downarrow g ) \simeq \operatorname{Sing}_{\bullet }(f) \downarrow \operatorname{Sing}_{\bullet }(g), \]

where the right hand side is the comma construction of Definition 5.1.5.1.

Remark 5.1.5.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ and $G^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ be the opposite morphisms. Then we have a canonical isomorphism of simplicial sets

\[ (F \downarrow G)^{\operatorname{op}} \simeq (G^{\operatorname{op}} \downarrow F^{\operatorname{op}} ). \]

We will be particularly interested in the special case of Definition 5.1.5.1.

Notation 5.1.5.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $X$ be a vertex of $\operatorname{\mathcal{D}}$, classified by a morphism of simplicial sets $G: \Delta ^0 \rightarrow \operatorname{\mathcal{D}}$. In this case, we denote the comma constructions $(F \downarrow G)$ and $(G \downarrow F)$ by $(F \downarrow X)$ and $(X \downarrow F)$, respectively. In the special case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{D}}$ and $F = \operatorname{id}_{\operatorname{\mathcal{C}}}$ is the identity morphism, we will also denote $(F \downarrow G)$ and $(G \downarrow F)$ by $(\operatorname{\mathcal{C}}\downarrow X)$ and $(X \downarrow \operatorname{\mathcal{C}})$, respectively.

Remark 5.1.5.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X$ be a vertex of $\operatorname{\mathcal{C}}$. For any simplicial set $K$, we have canonical isomorphisms of simplicial sets

\[ \operatorname{Fun}(K, (\operatorname{\mathcal{C}}\downarrow X) ) \simeq \operatorname{Fun}_{ \{ x\} / }( K \diamond \{ x\} , \operatorname{\mathcal{C}}) \quad \quad \operatorname{Fun}(K, (X \downarrow \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}_{ \{ x\} / }( \{ x\} \diamond K, \operatorname{\mathcal{C}}), \]

where $K \diamond \{ x\} $ and $\{ x\} \diamond K$ denote the simplicial sets defined in Notation 4.5.5.3. Restricting to vertices, we obtain bijections

\[ \{ \textnormal{Morphisms $K \rightarrow (\operatorname{\mathcal{C}}\downarrow X)$} \} \simeq \{ \textnormal{Morphisms $f: (K \diamond \{ x\} ) \rightarrow \operatorname{\mathcal{C}}$ with $f(x) = X$} \} \]
\[ \{ \textnormal{Morphisms $K \rightarrow (X \downarrow \operatorname{\mathcal{C}})$} \} \simeq \{ \textnormal{Morphisms $f: (\{ x\} \diamond K) \rightarrow \operatorname{\mathcal{C}}$ with $f(x) = X$} \} . \]

Example 5.1.5.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we identify with morphisms of simplicial sets $X,Y: \Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 5.1.1.1 can be identified with the comma construction

\[ (X \downarrow \operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} = (X \downarrow Y) = \{ X\} \times _{ \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{C}}\downarrow Y). \]

The following result is a relative version of Proposition 5.1.1.8:

Proposition 5.1.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then the projection map $(X \downarrow \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a left fibration and the projection map $(\operatorname{\mathcal{C}}\downarrow X) \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. We will prove the second assertion; the first follows by a similar argument. Let $A \hookrightarrow B$ be a right anodyne morphism of simplicial sets; we wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & (\operatorname{\mathcal{C}}\downarrow X) \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]

admits a solution. Unwinding the definitions, we are reduced to showing that a map of simplicial sets

\[ \sigma _0: B \coprod _{A} (A \diamond \{ X\} ) \rightarrow \operatorname{\mathcal{C}} \]

can be extended to a map $\sigma : B \diamond \{ X\} \rightarrow \operatorname{\mathcal{C}}$ (see Notation 4.5.5.3). By virtue of Lemma 4.5.6.2, it will suffice to show that the inclusion map

\[ \iota : B \coprod _{A} ( A \diamond \{ X\} ) \hookrightarrow B \diamond \{ X\} \]

is a categorical equivalence of simplicial sets, which follows from Corollary 4.5.5.12. $\square$

Corollary 5.1.5.11. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category containing an object $X$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then the projection map $(X \downarrow F) \rightarrow \operatorname{\mathcal{C}}$ is a left fibration, and the projection map $(F \downarrow X) \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. Unwinding the definition, we have pullback diagrams

\[ \xymatrix@R =50pt@C=50pt{ (X \downarrow F) \ar [r] \ar [d] & (X \downarrow \operatorname{\mathcal{D}}) \ar [d] & (F \downarrow X) \ar [r] \ar [d] & (\operatorname{\mathcal{D}}\downarrow X) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}& \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}. } \]

The desired result now follows by combining Proposition 5.1.5.10 with Remark 4.2.1.9. $\square$

Recall that, if $\operatorname{\mathcal{C}}$ is an ordinary category containing an object $X$, then the slice and coslice categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$ can be identified with the comma categories $(\operatorname{\mathcal{C}}\downarrow X)$ and $(X \downarrow \operatorname{\mathcal{C}})$, respectively (Remark 4.3.1.6). Our next goal is to establish a similar result in the setting of $\infty $-categories. Here the situation is a bit more subtle: if $X$ is an object of an $\infty $-category $\operatorname{\mathcal{C}}$, then the simplicial sets $\operatorname{\mathcal{C}}_{/X}$ and $( \operatorname{\mathcal{C}}\downarrow X)$ are generally not isomorphic. However, we will show that they are nevertheless equivalent as $\infty $-categories.

Construction 5.1.5.12. For each $n \geq 0$, let

\[ c^{\mathrm{L}}_{n}: \Delta ^1 \times \Delta ^{n} \simeq \Delta ^{0} \times \Delta ^1 \times \Delta ^{n} \rightarrow \Delta ^{0} \star \Delta ^{n} \simeq \Delta ^{n+1} \]
\[ c^{\mathrm{R}}_{n}: \Delta ^{n} \times \Delta ^{1} \simeq \Delta ^{n} \times \Delta ^1 \times \Delta ^{0} \rightarrow \Delta ^{n} \star \Delta ^{0} \simeq \Delta ^{n+1} \]

denote the collapse morphisms given by Construction 4.5.5.1, given concretely on vertices by the formulae

\[ c^{\mathrm{L}}_{n}(i,j) = \begin{cases} 0 & \textnormal{ if } i=0 \\ j+1 & \textnormal{ if } i=1 \end{cases} \quad \quad c^{\mathrm{R}}_{n}(j,i) = \begin{cases} j & \textnormal{ if } i = 0 \\ n+1 & \textnormal{ if } i = 1.\end{cases} \]

Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a vertex $X$, and let $\sigma $ be an $n$-simplex of the coslice simplicial set $\operatorname{\mathcal{C}}_{X/}$, which we identify with a morphism of simplicial sets $\overline{\sigma }: \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\sigma }(0)=X$. The composite map $\Delta ^{1} \times \Delta ^{n} \xrightarrow { c^{\mathrm{L}}_{n} } \Delta ^{n+1} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}$ can be identified with an $n$-simplex of the comma construction $(X \downarrow \operatorname{\mathcal{C}})$ which we will denote by $\iota ^{\mathrm{L}}_{X}(\sigma )$. The construction $\sigma \mapsto \iota ^{\mathrm{L}}_{X}(\sigma )$ determines a morphism of simplicial sets $\iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \rightarrow (X \downarrow \operatorname{\mathcal{C}})$, which we will refer to as the left-pinch inclusion map.

Similarly, let $\tau $ be an $n$-simplex of the slice simplicial set $\operatorname{\mathcal{C}}_{/X}$, which we identify with a morphism of simplicial sets $\overline{\tau }: \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\tau }(n+1) = X$. The composite map $\Delta ^ n \times \Delta ^1 \xrightarrow { c^{\mathrm{R}}_{n} } \Delta ^{n+1} \xrightarrow { \overline{\tau } } \operatorname{\mathcal{C}}$ can be identified with an $n$-simplex of the comma construction $(\operatorname{\mathcal{C}}\downarrow X)$ which we will denote by $\iota ^{\mathrm{R}}_{Y}(\tau )$. The construction $\sigma \mapsto \iota ^{\mathrm{R}}_{X}(\tau )$ determines a morphism of simplicial sets $\iota ^{\mathrm{R}}_{X}: \operatorname{\mathcal{C}}_{/X} \rightarrow (\operatorname{\mathcal{C}}\downarrow X)$, which we will refer to as the right-pinch inclusion map.

Remark 5.1.5.13. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a vertex $X$. Then the pinch inclusion maps

\[ \iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \rightarrow (X \downarrow \operatorname{\mathcal{C}}) \quad \quad \iota ^{\mathrm{R}}_{X}: \operatorname{\mathcal{C}}_{/X} \rightarrow (\operatorname{\mathcal{C}}\downarrow X) \]

are monomorphisms of simplicial sets: this follows immediately from surjectivity of the collapse morphisms

\[ c_{n}^{\mathrm{L}}: \Delta ^1 \times \Delta ^{n} \twoheadrightarrow \Delta ^{n+1} \quad \quad c_{n}^{\mathrm{R}}: \Delta ^{n} \times \Delta ^{1} \twoheadrightarrow \Delta ^{n+1}. \]

Theorem 5.1.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then the pinch inclusion maps

\[ \iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow (X \downarrow \operatorname{\mathcal{C}}) \quad \quad \iota ^{\mathrm{R}}_{X}: \operatorname{\mathcal{C}}_{/X} \hookrightarrow (\operatorname{\mathcal{C}}\downarrow X) \]

are equivalences of $\infty $-categories.

The proof of Theorem 5.1.5.14 will make use of the following:

Lemma 5.1.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $g: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram indexed by a simplicial set $A$. Suppose we are given a pair of diagrams $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}_{g/}$ indexed by a simplicial set $B$, which we identify with diagrams $f'_0, f'_1: (A \star B) \rightarrow \operatorname{\mathcal{C}}$ satisfying $f'_0|_{A} = g = f'_{1}|_{A}$. The following conditions are equivalent:

$(1)$

The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the diagram $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}}_{g/} )$.

$(2)$

The diagrams $f'_0$ and $f'_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{ A / }(A \star B, \operatorname{\mathcal{C}})$.

Proof. Choose a categorical mapping cylinder

\[ B \coprod B \xrightarrow {(s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]

for the simplicial set $B$ (Definition 5.1.4.3). Using Corollary 4.5.5.8, we deduce that the resulting diagram

\[ (A \star B) \coprod _{A} (A \star B) \xrightarrow {(s'_0, s'_1)} A \star \overline{B} \xrightarrow {\pi '} A \star B \]

is a categorical mapping cylinder for the join $A \star B$ relative to $A$. Using the criterion of Corollary 5.1.4.11, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

There exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}_{g/}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.

$(2')$

There exists a diagram $\overline{f}': A \star \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f'_0 = \overline{f}' \circ s'_0$ and $f'_1 = \overline{f}' \circ s'_1$.

The equivalence of $(1')$ and $(2')$ now follows immediately from the universal property of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{g/}$. $\square$

Proof of Theorem 5.1.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. We will show that the left-pinch inclusion morphism $\iota ^{\mathrm{L}}_{X}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow (X \downarrow \operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories; the analogous statement for $\iota ^{\mathrm{R}}_{X}$ follows by a similar argument. Let $K$ be a simplicial set and

\[ \rho : \pi _0( \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{X/})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(K, X \downarrow \operatorname{\mathcal{C}})^{\simeq } ) \]

be the map induced by composition with $\iota ^{\mathrm{L}}_{X}$; we wish to show that $\rho $ is a bijection. Using Lemma 5.1.5.15 and Remark 5.1.5.8, we can identify $\rho $ with the map

\[ \pi _0( \operatorname{Fun}_{\{ x\} / }( \{ x\} \star K, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}_{ \{ x\} }( \{ x\} \diamond K, \operatorname{\mathcal{C}})^{\simeq } ) \]

given by composition with the comparison map $c: \{ x\} \diamond K \rightarrow \{ x\} \star K$. We will complete the proof by showing that composition with $c$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{ \{ x\} / }( \{ x\} \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ \{ x\} }( \{ x\} \diamond K, \operatorname{\mathcal{C}})$. This follows by applying Corollary 4.5.4.4 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \{ x\} \star K, \operatorname{\mathcal{C}}) \ar [r]^-{\circ c} \ar [d] & \operatorname{Fun}( \{ x\} \diamond K, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \{ x\} , \operatorname{\mathcal{C}}) \ar@ {=}[r] & \operatorname{Fun}( \{ x\} , \operatorname{\mathcal{C}}); } \]

here the vertical maps are isofibrations (Corollary 4.4.5.3) and the upper horizontal map is an equivalence of $\infty $-categories by Proposition 4.5.2.8 (and the fact that $c$ is a categorical equivalence, by Theorem 4.5.5.7). $\square$