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5.2.3 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 5.2.3.1. 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 5.2.3.1 from a more general statement (Theorem 5.2.3.8), which we prove at the end of this section.

Remark 5.2.3.2. Proposition 5.2.3.1 has an interpretation in the language of model categories. Let us regard the category $\operatorname{Set_{\Delta }}$ of simplicial sets as equipped with the Joyal model structure of Remark . Conditions $(2)$ and $(3)$ of Proposition 5.2.3.1 are equivalent to the requirement that the morphisms $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ are homotopic with respect to the Joyal model structure (in the sense of Definition ). Proposition 5.2.3.1 asserts that this is equivalent to the requirement that $f_0$ and $f_1$ are naturally isomorphic (in the sense of Definition 4.4.4.1).

Let us introduce a bit of terminology which is useful for exploiting Proposition 5.2.3.1.

Definition 5.2.3.3. 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 $B$.

Remark 5.2.3.4. In the situation of Definition 5.2.3.3, condition $(2)$ is equivalent to the requirement that the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{i} \ar [d]^{i} & B \ar [d]^{s_0} \\ B \ar [r]^-{s_1} & \overline{B} } \]

commutes and is a pullback square (note that the morphisms $s_0$ and $s_1$ are automatically monomorphisms, since they are left inverse to the map $\pi : \overline{B} \rightarrow B$).

Remark 5.2.3.5. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets, and let $(\operatorname{id}_ B, \operatorname{id}_ B): (B \coprod _{A} B) \rightarrow B$ be the fold map. Unwinding the definitions, we see that a categorical mapping cylinder for $B$ relative to $A$ can be identified with a factorization of $(\operatorname{id}_ B, \operatorname{id}_ B)$ as a composition

\[ B \coprod _{A} B \xrightarrow {\iota } \overline{B} \xrightarrow {\pi } B, \]

where $\iota $ is a monomorphism of simplicial sets and $\pi $ is a categorical equivalence. Such factorizations always exist: by virtue of Exercise 3.1.6.11, we can even arrange that $\pi $ is a trivial Kan fibration of simplicial sets (hence a categorical equivalence by virtue of Proposition 4.5.2.9).

Example 5.2.3.6. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets, and let $Q$ be a 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 4.5.3.7). 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 5.2.3.7. 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 5.2.3.3). 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 5.2.3.1, and the implication $(b) \Rightarrow (a)$ from the implication $(2) \Rightarrow (1)$ of Proposition 5.2.3.1. $\square$

We will deduce Proposition 5.2.3.1 from a more general relative statement.

Theorem 5.2.3.8. 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 4.4.3.10, 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 5.2.3.6. 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 $(\operatorname{id}_ B, \operatorname{id}_ B): (\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ 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 3.1.6.11, 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 4.5.2.9), so the morphism $j$ is a categorical equivalence (Remark 4.5.2.6). 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 5.2.3.9. 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 5.2.3.8 in the special case where $S = \Delta ^{0}$. $\square$

Proof of Proposition 5.2.3.1. Apply Corollary 5.2.3.9 in the special case $A = \emptyset $. $\square$

For later use, we record a relative version of Corollary 5.2.3.7.

Corollary 5.2.3.10. 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 5.2.3.8, and the implication $(b) \Rightarrow (a)$ from the implication $(2) \Rightarrow (1)$ of Theorem 5.2.3.8. $\square$

Corollary 5.2.3.11. 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 5.2.3.10 in the special case $S = \Delta ^{0}$. $\square$