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8.4.1 Retracts in $\infty $-Categories

The notion of retract has an obvious counterpart in the setting of $\infty $-categories.

Definition 8.4.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. We say that an object $Y \in \operatorname{\mathcal{C}}$ is a retract of $X$ if there exist morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ for which the identity morphism $\operatorname{id}_{Y}$ is a composition of $i$ and $r$, in the sense of Definition 1.3.4.1.

Remark 8.4.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then an object $Y \in \operatorname{\mathcal{C}}$ is a retract of $X$ (in the sense of Definition 8.4.1.1) if and only if it is a retract of $X$ when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Remark 8.4.1.3. Let $\operatorname{\mathcal{C}}$ be a category containing an object $X$. Then an object $Y \in \operatorname{\mathcal{C}}$ is retract of $X$ if and only if it is a retract of $X$ when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 8.4.1.1). Consequently, Definition 8.4.1.1 can be viewed as a generalization of the classical notion of retract.

Example 8.4.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. If an object $Y \in \operatorname{\mathcal{C}}$ is isomorphic to $X$, then $Y$ is a retract of $X$. In particular, the object $X$ is a retract of itself.

Remark 8.4.1.5 (Transitivity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. If $Y$ is a retract of $X$ and $Z$ is a retract of $Y$, then $Z$ is a retract of $X$. To prove this, choose morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ satisfying $[r] \circ [i] = [ \operatorname{id}_{Y} ]$, and morphisms $j: Z \rightarrow Y$ and $s: Y \rightarrow Z$ satisfying $[s] \circ [j] = [ \operatorname{id}_{Z} ]$. We then compute

\[ ([s] \circ [r]) \circ ([i] \circ [j]) = [s] \circ ([r] \circ [i]) \circ [j] = [s] \circ [\operatorname{id}_{Y}] \circ [j] = [s] \circ [j] = [\operatorname{id}_{Z}], \]

so that $Z$ is a retract of $X$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

In practice, many important properties of an object $X$ of an $\infty $-category $\operatorname{\mathcal{C}}$ are inherited by any retract of $X$. We record a few examples of this phenomenon which will be useful later.

Proposition 8.4.1.6 (Retracts of Isomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: X \rightarrow X'$ and $g: Y \rightarrow Y'$. Suppose that $g$ is a retract of $f$ (when regarded as objects of the arrow $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$). If $f$ is an isomorphism, then $g$ is also an isomorphism.

Proof. By virtue of Remark 8.4.1.2, we may assume that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. Choose a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ Y \ar [d]^{g} \ar [r]^-{ i } & X \ar [r]^-{r} \ar [d]^{f} & Y \ar [d]^{g} \\ Y' \ar [r]^-{i' } & X' \ar [r]^-{r'} & Y', } \]

where the horizontal compositions are the identity morphisms $\operatorname{id}_{Y}$ and $\operatorname{id}_{Y'}$, respectively. If $f$ is an isomorphism, then $g$ is also an isomorphism, with invrse given by the composition $Y' \xrightarrow {i'} X' \xrightarrow { f^{-1} } X \xrightarrow {r} Y$. This follows from the calculations

\[ g \circ r \circ f^{-1} \circ i' = r' \circ f \circ f^{-1} \circ i' = r' \circ \operatorname{id}_{X'} \circ i' = r' \circ i' = \operatorname{id}_{Y'} \]

\[ r \circ f^{-1} \circ i' \circ g = r \circ f^{-1} \circ f \circ i = r \circ \operatorname{id}_{X} \circ i = r \circ i = \operatorname{id}_{Y}. \]
$\square$

Proposition 8.4.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that every object $Y \in \operatorname{\mathcal{C}}$ is a retract of some object $X \in \operatorname{\mathcal{C}}^{0}$. Then $F$ is left and right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. We will show that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$; the assertion that $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ follows by a similar argument. Choose a regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Using Corollary 8.2.5.12, we can choose a fully faithful functor $\operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$, where $\widehat{\operatorname{\mathcal{D}}}$ admits $\kappa $-small colimits. By virtue of Remark 7.3.1.13, we can replace $\operatorname{\mathcal{D}}$ by $\widehat{\operatorname{\mathcal{D}}}$ and thereby reduce to proving Proposition 8.4.1.7 in the special case where $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits.

Set $F^{0} = F|_{ \operatorname{\mathcal{C}}^{0} }$. Using Proposition 7.6.7.12, we can extend $F^{0}$ to a functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Invoking the universal mapping property of Corollary 7.3.6.9, we deduce that there is a natural transformation $\alpha : F' \rightarrow F$ which restricts to the identity transformation from $F^{0}$ to itself. The natural transformation $\alpha $ carries each object $Y \in \operatorname{\mathcal{C}}$ to a morphism $\alpha _{Y}: F'(Y) \rightarrow F(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$. By assumption, the object $Y$ is a retract of some object $X \in \operatorname{\mathcal{C}}^{0}$. It follows that $\alpha _ Y$ is a retract of the morphism $\alpha _{X} = \operatorname{id}_{ F(X) }$, and is therefore an isomorphism (Proposition 8.4.1.6). Invoking Theorem 4.4.4.4, we deduce that the natural transformation $\alpha $ is an isomorphism, so that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ by virtue of Remark 7.3.3.14. $\square$

Proposition 8.4.1.7 immediately implies the following stronger version of Proposition 7.3.3.6:

Corollary 8.4.1.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $X \in \operatorname{\mathcal{C}}$. If $Y \in \operatorname{\mathcal{C}}$ is a retract of $X$, then $F$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $Y$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is spanned by $\operatorname{\mathcal{C}}^{0}$ together with the objects $X$ and $Y$. By virtue of Proposition 7.3.7.6, we can further assume that $\operatorname{\mathcal{C}}^{0}$ contains the object $X$. In this case, Proposition 8.4.1.7 implies that the functors $F$ and $U \circ F$ are left Kan extended from $\operatorname{\mathcal{C}}^{0}$, so that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ by virtue of Remark 7.3.3.17. $\square$

Corollary 8.4.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Suppose that $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$. Then any functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is a retract of $F$ (in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$) is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$.

Proof. Let $\operatorname{ev}: \operatorname{\mathcal{C}}\times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ denote the evaluation functor. By virtue of Remark 7.3.3.3, the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $C \in \operatorname{\mathcal{C}}$ if and only if the functor $\operatorname{ev}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ at the object $(C,F)$. If this condition is satisfied, then Corollary 8.4.1.8 guarantees that $\operatorname{ev}$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ at the object $(C,G)$, so that $G$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$. The desired result now follows by allowing the object $C \in \operatorname{\mathcal{C}}$ to vary. $\square$

Corollary 8.4.1.10. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories, let $K$ be a simplicial set, and suppose we are given a pair of diagrams $f,g: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. If $f$ is a $U$-colimit diagram and $g$ is a retract of $f$ (in the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$), then $g$ is also a $U$-colimit diagram.

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. Using Remark 4.3.6.7, we see that the induced map $K^{\triangleright } \hookrightarrow \operatorname{\mathcal{K}}^{\triangleright }$ is also inner anodyne. We may therefore extend $f$ and $g$ to functors $F,G: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. Since the restriction functor $\operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$ is a trivial Kan fibration (Proposition 1.4.7.6), it follows that $G$ is a retract of $F$. By virtue of Corollary 7.2.2.2, we can replace $K$ by $\operatorname{\mathcal{K}}$ and thereby reduce to proving Corollary 8.4.1.10 in the special case where $K$ is an $\infty $-category. In this case, the desired result is a special case of Corollary 8.4.1.9 (see Example 7.3.3.8). $\square$

Corollary 8.4.1.11. Let $K$ be a simplicial set and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves $K$-indexed colimits. If $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a retract of $F$ (in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), then $G$ also preserves $K$-indexed colimits.

Proposition 8.4.1.12. Let $\operatorname{\mathcal{X}}$ and $\operatorname{\mathcal{Y}}$ be $\infty $-categories and let $\kappa $ be an uncountable cardinal. Suppose that $\operatorname{\mathcal{Y}}$ is a retract of $\operatorname{\mathcal{X}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$. If $\operatorname{\mathcal{X}}$ is essentially $\kappa $-small, then $\operatorname{\mathcal{Y}}$ is also essentially $\kappa $-small.

Proof. By virtue of Proposition 5.4.6.12, we may assume that the $\infty $-categories $\operatorname{\mathcal{X}}$ and $\operatorname{\mathcal{Y}}$ are minimal, so that $\operatorname{\mathcal{X}}$ is a $\kappa $-small simplicial set (Corollary 5.4.6.9). Choose functors $i: \operatorname{\mathcal{Y}}\rightarrow \operatorname{\mathcal{X}}$ and $r: \operatorname{\mathcal{X}}\rightarrow \operatorname{\mathcal{Y}}$ such that the composition $(r \circ i): \operatorname{\mathcal{Y}}\rightarrow \operatorname{\mathcal{Y}}$ is isomorphic to the identity functor. Then $r \circ i$ is an equivalence of $\infty $-categories. Since $\operatorname{\mathcal{Y}}$ is minimal, it follows that $r \circ i$ is an isomorphism of simplicial sets (Proposition 5.4.6.10). In particular, the functor $i: \operatorname{\mathcal{Y}}\rightarrow \operatorname{\mathcal{X}}$ is a monomorphism of simplicial sets. It follows that $\operatorname{\mathcal{Y}}$ is $\kappa $-small (Remark 5.4.4.8), and therefore essentially $\kappa $-small. $\square$

Remark 8.4.1.13. In the situation of Proposition 8.4.1.12, suppose that the $\infty $-category $\operatorname{\mathcal{X}}$ is a Kan complex. Then $\operatorname{\mathcal{Y}}$ is also a Kan complex, and is therefore a retract of $X$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. To prove this, it will suffice to show that every morphism $f: Y \rightarrow Y'$ in the $\infty $-category $\operatorname{\mathcal{Y}}$ is an isomorphism (Proposition 4.4.2.1). Since $\operatorname{\mathcal{X}}$ is a Kan complex, the morphism $i(f): i(Y) \rightarrow i(Y')$ is an isomorphism in $\operatorname{\mathcal{X}}$ (Proposition 1.3.6.10). It follows that $(r \circ i)(f)$ is an isomorphism in $\operatorname{\mathcal{Y}}$ (Remark 1.4.1.6). Since $f$ is isomorphic to $(r \circ i)(f)$ (as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{Y}})$), it is also an isomorphism (Example 4.4.1.13).

Corollary 8.4.1.14. Let $X$ and $Y$ be Kan complexes and let $\kappa $ be an uncountable cardinal. Suppose that $Y$ is a retract of $X$ in the $\infty $-category $\operatorname{\mathcal{S}}$. If $X$ is essentially $\kappa $-small, then $Y$ is essentially $\kappa $-small.

Warning 8.4.1.15. In the statement of Corollary 8.4.1.14, the uncountability assumption on $\kappa $ cannot be omitted. That is, if $X$ is a Kan complex for which there exists a weak homotopy equivalence $K \rightarrow X$ for a finite simplicial set $K$, then a retract of $X$ need not inherit the same property. See §.

We now make Definition 8.4.1.1 slightly more explicit.

Definition 8.4.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A retraction diagram in $\operatorname{\mathcal{C}}$ is a $2$-simplex $\sigma : \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ for which the “long” face $d_{1}(\sigma )$ is an identity morphism of $\operatorname{\mathcal{C}}$. In this case, we indicate $\sigma $ by a diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y, } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$, and we say that $\sigma $ exhibits $Y$ as a retract of $X$.

Remark 8.4.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then an object $Y \in \operatorname{\mathcal{C}}$ is a retract of $X$ (in the sense of Definition 8.4.1.1) if and only if there exists a retraction diagram which exhibits $Y$ as a retract of $X$ (in the sense of Definition 8.4.1.16).

Warning 8.4.1.18. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then a retraction diagram in $\operatorname{\mathcal{C}}$ can be identified with a pair of morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ satisfying the condition $r \circ i = \operatorname{id}_{Y}$. Beware that, if $\operatorname{\mathcal{C}}$ is a general $\infty $-category, then a retraction diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y } \]

generally cannot be recovered (even up to isomorphism) from the morphisms $i$ and $r$ alone: one also needs a homotopy which witnesses the identity $[r] \circ [i] = [ \operatorname{id}_{Y} ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Remark 8.4.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ is a retraction diagram if and only if it is a retraction diagram when viewed as an object of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Consequently, if $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then $Y$ is a retract of $X$ in $\operatorname{\mathcal{C}}$ if and only if it is a retract of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Our goal in this section is carry out an $\infty $-categorical analogue of Exercise 8.4.0.2.

Notation 8.4.1.20. Let $\operatorname{Ret}$ denote the category introduced in Construction 8.4.0.1. By construction, the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Ret})$ contains a retraction diagram $\sigma : \Delta ^2 \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$, which we depict as

\[ \xymatrix@R =50pt@C=50pt{ & \widetilde{X} \ar [dr]^{ \widetilde{r} } & \\ \widetilde{Y} \ar [ur]^{\widetilde{i}} \ar [rr]^{ \operatorname{id}} & & \widetilde{Y}. } \]

We let $\mathcal{R}$ denote the image of $\sigma $, which we regard as a simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{Ret})$.

Remark 8.4.1.21. In the situation of Notation 8.4.1.20, the map $\sigma : \Delta ^2 \twoheadrightarrow \mathcal{R}$ is an epimorphism of simplicial sets, which fits into a pushout square

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \ar [r] \ar [d] & \Delta ^0 \ar [d] \\ \Delta ^2 \ar [r]^-{\sigma } & \mathcal{R}. } \]

It follows that, for every $\infty $-category $\operatorname{\mathcal{C}}$, composition with $\sigma $ induces a bijection from $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \mathcal{R}, \operatorname{\mathcal{C}})$ to the set of retraction diagrams in $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.4.1.16).

Remark 8.4.1.22. Let $\sigma : \Delta ^2 \twoheadrightarrow \mathcal{R}$ be the epimorphism of Notation 8.4.1.20. For every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $\sigma $ induces a fully faithful functor $\operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$, whose essential image is the full subcategory $\operatorname{Fun}'( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ u} & & Y' } \]

where $u$ is an isomorphism. This follows by applying Corollary 4.5.2.23 to the pullback square

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}'( \Delta ^2, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Isom}(\operatorname{\mathcal{C}}), } \]

since the vertical maps are isofibrations (Corollary 4.4.5.3) and the lower horizontal map is an equivalence of $\infty $-categories by virtue of Corollary 4.5.3.13.

Our main result can now be stated as follows:

Proposition 8.4.1.23. The inclusion map $\mathcal{R} \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{Ret})$ is an inner anodyne morphism of simplicial sets.

Corollary 8.4.1.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the inclusion map $\mathcal{R} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$ induces a trivial Kan fibration

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \{ 0 < 2 \} ), \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]

In particular, every retraction diagram in $\operatorname{\mathcal{C}}$ can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$, which is uniquely determined up to isomorphism.

Corollary 8.4.1.26. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the retraction diagram of Notation 8.4.1.20 induces a fully faithful functor $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$, whose essential image is spanned by those diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ u} & & Y' } \]

where $u$ is an isomorphism.

Corollary 8.4.1.27. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a filtered category $\operatorname{\mathcal{I}}$. Suppose that each $\operatorname{\mathcal{C}}_{i}$ is an $\infty $-category. Then the tautological map

\[ \theta : \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ) \]

is an equivalence of $\infty $-categories.

Proof. The morphism $\theta $ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ) \ar [d] \\ \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}_ i ) \ar [r]^-{\theta '} & \operatorname{Fun}( \mathcal{R}, \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ), } \]

where the vertical maps are trivial Kan fibrations (Corollary 8.4.1.26). It will therefore suffice to show that $\theta '$ is an equivalence of $\infty $-categories. In fact, $\theta '$ is an isomorphism of simplicial sets, since the simplicial set $\mathcal{R}$ is finite (Corollary 3.5.1.10). $\square$

The proof of Proposition 8.4.1.23 will require the following:

Lemma 8.4.1.28 (Sparse Horns). Let $n \geq 0$ be an integer and let $S$ be a subset of $[n] = \{ 0 < 1 < \cdots < n \} $. Let $K \subseteq \Delta ^ n$ be the simplicial subset spanned by those nondegenerate simplices which do not contain every element of $S$. Suppose that there exist $0 \leq i < j < k \leq n$ such that $i,k \in S$, $j \notin S$. Then the inclusion $K \hookrightarrow \Delta ^ n$ is inner anodyne.

Example 8.4.1.29. In the situation of Lemma 8.4.1.28, suppose that $S = [n] \setminus \{ j\} $ for some $0 \leq j \leq n$. Then $K$ is the horn $\Lambda ^{n}_{j} \subseteq \Delta ^ n$. The hypothesis of Lemma 8.4.1.28 guarantees that $K$ is an inner horn, so that the inclusion map $K \hookrightarrow \Delta ^ n$ is inner anodyne by definition.

Proof of Lemma 8.4.1.28. Let $P$ denote the collection of all subsets $S' \subseteq [n]$ which contain $S \cup \{ j\} $. Choose a linear ordering

\[ \{ S(1) \leq \cdots \leq S(c) \} \]

of $P$ with the property that if $S(a) \subseteq S(b)$, then $a \leq b$. Let For $0 \leq b \leq c$, let $K(b) \subseteq \Delta ^ n$ denote the union of $K$ with the faces $\{ \operatorname{N}_{\bullet }( S(a) ) \subseteq \Delta ^ n \} _{ 1 \leq a \leq b}$. We then have inclusion maps

\[ K = K(0) \subseteq K(1) \subseteq K(2) \subseteq \cdots \subseteq K(c-1) \subseteq K(c) = \Delta ^ n. \]

It will therefore suffice to show that, for every positive integer $b \leq c$, the inclusion map $K(b-1) \hookrightarrow K(b)$ is inner anodyne.

Let us identify $\operatorname{N}_{\bullet }( S(b) )$ with the image of a nondegenerate simplex $\sigma : \Delta ^ m \hookrightarrow \Delta ^ n$. Let $L \subseteq \Delta ^ m$ be the inverse image $\sigma ^{-1}( S(b-1) )$, so that we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ L \ar [r] \ar [d] & \Delta ^ m \ar [d]^{\sigma } \\ S(b-1) \ar [r] & S(b). } \]

It will therefore suffice to show that the inclusion map $L \subseteq \Delta ^ m$ is inner anodyne. Because $S(b)$ contains the integers $i < j < k$, we can write $j = \sigma (\overline{j})$ for some $0 < \overline{j} < n$. We conclude by observing that $L$ can be identified with the inner horn $\Lambda ^{m}_{\overline{j} } \subseteq \Delta ^ m$. $\square$

Proof of Proposition 8.4.1.23. Let $\tau $ be a nondegenerate $m$-simplex of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Ret})$. We define the weight $w(\tau )$. to be the cardinality of the set $\{ i \in [m]: \tau (i) = \widetilde{X} \} $. Note that, if $\tau '$ is any nondegenerate facet of $\tau $, then $w( \tau ' ) \leq w( \tau )$. For $n \geq 1$, the collection of nondegenerate simplices of weight $\leq n$ span a simplicial subset $\mathcal{R}(n) \subseteq \operatorname{N}_{\bullet }( \operatorname{Ret})$. It follows that we can write $\operatorname{N}_{\bullet }( \operatorname{Ret})$ as the union of an increasing sequence

\[ \mathcal{R}(1) \hookrightarrow \mathcal{R}(2) \hookrightarrow \mathcal{R}(3) \hookrightarrow \cdots , \]

where $\mathcal{R}(1)$ coincides with the simplicial set $\mathcal{R}$ introduced in Notation 8.4.1.20. We will complete the proof by showing that, for each $n \geq 2$, the inclusion map $\mathcal{R}(n-1) \hookrightarrow \mathcal{R}(n)$ is inner anodyne.

Let $\sigma _{n}: \Delta ^{2n} \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$ denote the simplex corresponding to the diagram

\[ \widetilde{Y} \xrightarrow { \widetilde{i} } \widetilde{X} \xrightarrow {\widetilde{r}} \widetilde{Y} \xrightarrow {\widetilde{i}} \widetilde{X} \rightarrow \cdots \rightarrow \widetilde{X} \xrightarrow { \widetilde{r} } \widetilde{Y} \xrightarrow {\widetilde{i}} \widetilde{X} \xrightarrow {\widetilde{r}} \widetilde{Y}. \]

Note that $\sigma _{n}$ is a nondegenerate simplex of weight $n$, and therefore factors through $\mathcal{R}(n)$. Let $K \subseteq \Delta ^{2n}$ denote the inverse image $\sigma _{n}^{-1} \mathcal{R}(n-1)$, so that we have a commutative diagram of simplicial sets

8.26
\begin{equation} \begin{gathered}\label{equation:sparse-horn} \xymatrix@R =50pt@C=50pt{ K \ar [r] \ar [d] & \Delta ^{2n} \ar [d]^{ \sigma _ n } \\ \mathcal{R}(n-1) \ar [r] & \mathcal{R}(n). } \end{gathered} \end{equation}

Note that a nondegenerate simplex of $\Delta ^{2n}$ belongs to $K$ if and only if it does not contain $\operatorname{N}_{\bullet }( \{ 1 < 3 < \cdots < 2n-1 \} ) \subseteq \Delta ^{2n}$. Applying Lemma 8.4.1.28, we deduce that the inclusion map $K \hookrightarrow \Delta ^{2n}$ is inner anodyne. It will therefore suffice to show that the diagram (8.26) is a pushout square.

Let $\tau $ be an $m$-simplex of $\operatorname{N}_{\bullet }(\operatorname{Ret})$ which belongs to $\mathcal{R}(n)$, but does not belong to $\mathcal{R}(n-1)$. We wish to show that $\tau $ factors uniquely through $\sigma _{n}$. We first prove the existence of the desired factorization. For this, we may assume without loss of generality that $\tau $ is nondegenerate. Then $\tau $ has weight $n$, so we can write

\[ \{ i \in [m]: \tau (i) = X \} = \{ d_1 < d_2 < \cdots < d_ n \} . \]

Let $\alpha : \Delta ^ m \rightarrow \Delta ^{2n}$ be the unique morphism of simplices which is given on vertices by the formula $\alpha ( d_ i ) = 2i - 1$ for $1 \leq i \leq n$. We claim that $\tau = \sigma _{n} \circ \alpha $. Note that $\tau $ and $\sigma _ n \circ \alpha $ can both be regarded as functors from the linearly ordered set $[m]$ to the category $\operatorname{Ret}$. By construction, these functors coincide on objects. It will therefore suffice to show that, for $0 \leq j < j' < m$, the functors $\tau $ and $\sigma _{n} \circ \alpha $ determine the same element of $\operatorname{Hom}_{\operatorname{Ret}}( \tau (j), \tau (j') )$. If $\tau (j) = \widetilde{Y}$ or $\tau (j') = \widetilde{Y}$, this condition is automatic (since the set $\operatorname{Hom}_{\operatorname{Ret}}( \tau (j), \tau (j') )$ has only one element). We may therefore assume without loss of generality that $\tau (j) = \widetilde{X} = \tau (j')$: that is, we have $j = d_ i$ and $j' = d_{i'}$ for some $i < i'$. In this case, the functors $\tau $ and $\sigma _{n} \circ \alpha $ both carry the pair $(j < j')$ to the element $e \in \operatorname{Hom}_{\operatorname{Ret}}(X,X)$.

We now prove uniqueness. Suppose we are given a pair of maps $\alpha , \beta : \Delta ^{m} \rightarrow \Delta ^{2n}$ satisfying $\sigma _{n} \circ \alpha = \tau = \sigma _{n} \circ \beta $; we wish to show that $\alpha = \beta $. Suppose otherwise. Then there is some smallest integer $j \in [m]$ such that $\alpha (j) \neq \beta (j)$. Without loss of generality, we may assume that $\alpha (j) < \beta (j)$. Assume first that $\alpha (j)$ is odd. Since $\tau $ does not belong to $K$, $\alpha (j)$ is contained in the image of $\beta $; that is, we can write $\alpha (j) = \beta (i)$ for some $i < j$. Then minimality of $j$ then guarantees that $\alpha (i) = \alpha (j)$, so that $\sigma _{n} \circ \alpha $ carries the pair $(i < j)$ to the identity morphism $\operatorname{id}_{\widetilde{X}}$ in the category $\operatorname{Ret}$. Since $\sigma _{n} \circ \beta = \tau = \sigma _{n} \circ \alpha $, the morphism $\sigma _{n} \circ \beta $ also carries $(i < j)$ to the identity morphism $\operatorname{id}_{\widetilde{X}}$. It follows that $\beta (i) = \beta (j)$, contradicting our assumption that $\beta (i) = \alpha (j) < \beta (j)$.

We now treat the case where $\alpha (j)$ is even, so that $\tau (j) = (\sigma _{n} \circ \alpha )(j) = Y$. Using the equality $\sigma _{n} \circ \beta = \tau $, we deduce that $\beta (j)$ is also even. Since $\tau $ does not belong to $K$, the odd number $\beta (j)-1$ belongs to the image of $\beta $. We therefore have $\beta (j) - 1 = \beta (i)$ for some integer $i < j$. We then have

\[ \alpha (i) \leq \alpha (j) < \alpha (j) + 1 \leq \beta (j) - 1 = \beta (i), \]

contradicting the minimality of $j$. $\square$