4.6.3 Digression: Categorical Mapping Cylinders
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be diagrams in $\operatorname{\mathcal{C}}$ indexed by a simplicial set $B$. Recall that $f_0$ and $f_1$ are naturally isomorphic if they are isomorphic as objects of the diagram $\infty $-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$ (Definition 4.4.4.1). Our goal in this section is to establish a detection criterion for natural isomorphisms.
Proposition 4.6.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams. The following conditions are equivalent:
- $(1)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.
- $(2)$
There exists a factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod B \rightarrow B$ as a composition
\[ B \coprod B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]
where $\pi $ is a categorical equivalence, and a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
- $(3)$
For every factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod B \rightarrow B$ as a composition
\[ B \coprod B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]
where $s_0$ and $s_1$ have disjoint images, there exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
We will deduce Proposition 4.6.3.1 from a more general statement (Theorem 4.6.3.8), which we prove at the end of this section.
Let us introduce a bit of terminology which is useful for exploiting Proposition 4.6.3.1.
Definition 4.6.3.3. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. A categorical mapping cylinder for $B$ relative to $A$ is a simplicial set $\overline{B}$ equipped with a morphism $\pi : \overline{B} \rightarrow B$ together with a pair of sections $s_0, s_1: B \rightarrow \overline{B}$ having the following properties:
The morphism $\pi : \overline{B} \rightarrow B$ is a categorical equivalence of simplicial sets.
The morphisms $s_0, s_1: B \rightarrow \overline{B}$ satisfy $s_0 \circ i = s_1 \circ i$, and the induced map $(s_0, s_1): (B \coprod _{A} B) \rightarrow \overline{B}$ is a monomorphism.
If these conditions are satisfied in the special case $A = \emptyset $, we will simply refer to $\overline{B}$ (together with the morphisms $\pi $, $s_0$, and $s_1$) as a categorical mapping cylinder for $B$.
Example 4.6.3.6. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets, and let $Q$ be a contractible Kan complex containing vertices $x_0, x_1 \in Q$ with $x_0 \neq x_1$. Set $\overline{B} = A \coprod _{ (Q \times A) } (Q \times B)$. The commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ Q \times A \ar [r] \ar [d] & Q \times B \ar [d] \\ A \ar [r]^-{i} & B } \]
is a categorical pushout square (since the vertical maps are categorical equivalences), and therefore induces a categorical equivalence $\overline{\pi }: \overline{B} \rightarrow B$ (Proposition 4.5.4.11). Let $s_0: B \rightarrow \overline{B}$ be the section of $\pi $ given by the composition
\[ B \simeq \{ x_0 \} \times B \hookrightarrow Q \times B \rightarrow \overline{B}, \]
and define $s_1: B \rightarrow \overline{B}$ similarly. Then the quadruple $(\overline{B}, \pi , s_0, s_1)$ is a categorical mapping cylinder of $B$ relative to $A$.
Corollary 4.6.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by a simplicial set $B$. Let
\[ (B \coprod B) \xrightarrow {(s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
be a categorical mapping cylinder for $B$ (Definition 4.6.3.3). The following conditions are equivalent:
- $(a)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.
- $(b)$
There exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
In particular, condition $(b)$ does not depend on the choice of categorical mapping cylinder.
Proof.
The implication $(a) \Rightarrow (b)$ follows from the implication $(1) \Rightarrow (3)$ of Proposition 4.6.3.1, and the implication $(b) \Rightarrow (a)$ from the implication $(2) \Rightarrow (1)$ of Proposition 4.6.3.1.
$\square$
We will deduce Proposition 4.6.3.1 from a more general relative statement.
Theorem 4.6.3.8. Let $q: X \rightarrow S$ be an isofibration of simplicial sets, let $g: B \rightarrow S$ be a morphism of simplicial sets, and let $f_0, f_1: B \rightarrow X$ be morphisms satisfying $q \circ f_0 = g = q \circ f_1$. Let $A \subseteq B$ be a simplicial subset satisfying $f_0|_{A} = f_1|_{A}$. The following conditions are equivalent:
- $(1)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/ \, /S}(B, X)$ (see Proposition 4.1.4.6).
- $(2)$
There exists a factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]
where $\pi $ is a categorical equivalence and the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{ \overline{f} } & S } \]
admits a solution.
- $(3)$
For every factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]
where the map $(s_0, s_1): B \coprod _{A} B \rightarrow \overline{B}$ is a monomorphism, the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{ \overline{f} } & S } \]
admits a solution.
Proof.
By virtue of Corollary 4.4.3.15, condition $(1)$ is satisfied if and only if there exists a morphism of simplicial sets $u: Q \rightarrow \operatorname{Fun}_{A/\, /S}( B, X)$, where $Q$ is a contractible Kan complex, and a pair of vertices $x_0, x_1 \in Q$ satisfying $u(x_0) = f_0$ and $u(x_1) =f_1$. Moreover, we may assume (modifying $Q$ if necessary) that the vertices $x_0$ and $x_1$ are distinct. In this case, let $\overline{B} = A \coprod _{ (Q \times A) } (Q \times B)$ and let
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
be the categorical mapping cylinder described in Example 4.6.3.6. Unwinding the definitions, we see that morphisms $u: Q \rightarrow \operatorname{Fun}_{A/\, /S}( B, X)$ satisfying $u(x_0) = f_0$ and $u(x_1) = f_1$ can be identified with solutions to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur] & S. } \]
This proves that $(3) \Rightarrow (1) \Rightarrow (2)$.
We will complete the proof by showing that $(2) \Rightarrow (3)$. Assume that $(2)$ is satisfied, so that the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ factors as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
where $\pi $ is a categorical equivalence and there exists a morphism $\overline{f}: \overline{B} \rightarrow X$ for which the diagram
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar [ur]^{ \overline{f} } & S } \]
commutes. Using Exercise 3.1.7.11, we can factor $\pi $ as a composition
\[ \overline{B} \xrightarrow {j} \overline{B}' \xrightarrow {\pi '} B, \]
where $j$ is a monomorphism and $\pi '$ is a trivial Kan fibration. Then $\pi '$ is also a categorical equivalence (Proposition 4.5.3.11), so the morphism $j$ is a categorical equivalence (Remark 4.5.3.5). Our assumption that $q$ is an isofibration guarantees that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \overline{B} \ar [d]_{j} \ar [r]^-{ \overline{f} } & X \ar [d]^{q} \\ \overline{B}' \ar [r]^-{ g \circ \pi ' } \ar@ {-->}[ur]^{ \overline{f}' } & S } \]
admits a solution $\overline{f}': \overline{B}' \rightarrow X$.
We now show that condition $(3)$ is satisfied. Suppose that we are given another factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): (\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow {\iota } \overline{B}'' \xrightarrow {\pi ''} B, \]
where $\iota $ is a monomorphism. We wish to show that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ \iota } \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B}'' \ar [r]^-{ g \circ \pi ''} \ar@ {-->}[ur]^{\overline{f}''} & S } \]
admits a solution $\overline{f}'': \overline{B}'' \rightarrow X$. We first observe that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [r]^-{j \circ (s_0, s_1)} \ar [d]_{\iota } & \overline{B}' \ar [d]^{ \pi ' } \\ \overline{B}'' \ar [r]^-{\pi ''} \ar@ {-->}[ur]^{v} & B } \]
admits a solution $v: \overline{B}'' \rightarrow \overline{B}'$, since $\iota $ is a monomorphism and $\pi '$ is a trivial Kan fibration. We now conclude the proof by setting $\overline{f}'' = \overline{f}' \circ v$.
$\square$
Corollary 4.6.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by a simplicial set $B$, and let $A \subseteq B$ be a simplicial subset satisfying $f_0|_{A} = f_1|_{A}$. The following conditions are equivalent:
- $(1)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$.
- $(2)$
There exists a factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]
where $\pi $ is a categorical equivalence, and a morphism $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
- $(3)$
For every factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
where the map $(s_0, s_1): B \coprod _{A} B \rightarrow \overline{B}$ is a monomorphism, there exists a morphism $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
Proof.
Apply Theorem 4.6.3.8 in the special case where $S = \Delta ^{0}$.
$\square$
Proof of Proposition 4.6.3.1.
Apply Corollary 4.6.3.9 in the special case $A = \emptyset $.
$\square$
For later use, we record a relative version of Corollary 4.6.3.7.
Corollary 4.6.3.10. Let $q: X \rightarrow S$ be an isofibration of simplicial sets, let $g: B \rightarrow S$ be a morphism of simplicial sets, and let $f_0, f_1: B \rightarrow X$ be morphisms satisfying $q \circ f_0 = g = q \circ f_1$. Let $A$ be a simplicial subset of $B$ satisfying $f_0|_{A} = f_1|_{A}$, and let
\[ (B \coprod _{A} B) \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
be a categorical mapping cylinder of $B$ relative to $A$. The following conditions are equivalent:
- $(a)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/ \, /S}(B, X)$.
- $(b)$
The lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{ \overline{f} } & S } \]
admits a solution.
In particular, condition $(b)$ does not depend on the choice of categorical mapping cylinder.
Proof.
The implication $(a) \Rightarrow (b)$ follows from the implication $(1) \Rightarrow (3)$ of Theorem 4.6.3.8, and the implication $(b) \Rightarrow (a)$ from the implication $(2) \Rightarrow (1)$ of Theorem 4.6.3.8.
$\square$
Corollary 4.6.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by a simplicial set $B$. Let $A$ be a simplicial subset of $B$ satisfying $f_0|_{A} = f_1|_{A}$, and let
\[ (B \coprod _{A} B) \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
be a categorical mapping cylinder of $B$ relative to $A$. The following conditions are equivalent:
- $(a)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$.
- $(b)$
There exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
In particular, condition $(b)$ does not depend on the choice of categorical mapping cylinder.
Proof.
Apply Corollary 4.6.3.10 in the special case $S = \Delta ^{0}$.
$\square$
