Kerodon

Proof. Let $\sigma , \sigma ': \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be $n$-simplices of $\operatorname{\mathcal{C}}$ satisfying $F(\sigma ) = F(\sigma ')$; we wish to show that $\sigma = \sigma '$. Our proof proceeds by induction on $n$. Set $\tau = F(\sigma ) = F(\sigma ')$ and $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^ n }$, so that our inductive hypothesis guarantees that $\sigma _0 = \sigma '|_{ \operatorname{\partial \Delta }^ n}$.

Fix a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which is homotopy inverse to $F$, so that there exists a $2$-simplex

in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$, where $\alpha $ and $\beta $ are (mutually inverse) isomorphisms. Precomposing with the morphism $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$, we obtain a $2$-simplex

in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$. Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, Theorem 1.4.6.1 guarantees that we can lift (5.35) to a $2$-simplex

in the $\infty $-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})$. By construction, $\gamma $ is an isomorphism whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$ is an identity morphism. Invoking our assumption that $\operatorname{\mathcal{C}}$ is minimal, we deduce that $\sigma = \sigma '$. $\square$

Proof. Suppose that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Then there exists an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is $\kappa $-small. Since $\operatorname{\mathcal{C}}$ is minimal, the functor $F$ is a monomorphism of simplicial sets (Lemma 5.4.6.8), so that $\operatorname{\mathcal{C}}$ is also $\kappa $-small (Remark 5.4.4.8). $\square$

Proof. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse to $F$. It follows from Lemma 5.4.6.8 that $F$ and $G$ are monomorphisms of simplicial sets. We will complete the proof by showing that the composite map $(F \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is an epimorphism of simplicial sets (so that, in particular, $F$ is an epimorphism). Let $\sigma $ be an $n$-simplex of $\operatorname{\mathcal{D}}$; we wish to show that $\sigma $ belongs to the image of $F \circ G$. The proof proceeds by induction on $n$. Set $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^ n }$; our inductive hypothesis then guarantees that we can write $\sigma _0 = (F \circ G)( \tau _0 )$ for some morphism $\tau _0: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{D}}$.

Choose a $2$-simplex

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, where $\alpha $ and $\beta $ are isomorphisms. Precomposing with $\tau _0: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{D}}$, we obtain a $2$-simplex

in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}})$. Using Corollary 4.4.5.9, we can lift $\alpha (\tau _0)$ to an isomorphism $\widetilde{\alpha }: \sigma \rightarrow \tau $ in the $\infty $-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$. Since $\operatorname{\mathcal{D}}$ is an $\infty $-category, Theorem 1.4.6.1 guarantees that we can lift (5.36) to a $2$-simplex

in the $\infty $-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$. By construction, $\gamma $ is an isomorphism whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}})$ is an identity morphism. Our assumption that $\operatorname{\mathcal{D}}$ is minimal then guarantees that $\sigma = (F \circ G)(\tau )$ belongs to the image of $F \circ G$. $\square$

Proof. Injectivity is a restatement of Corollary 5.4.6.11, and surjectivity follows from Proposition 5.4.6.12. $\square$

Proof. By virtue of Proposition 5.4.6.12, we may assume that $\operatorname{\mathcal{C}}$ is a minimal $\infty $-category. In this case, the desired result follows by combining Corollary 5.4.6.9 with Remark 5.4.4.7. $\square$

Proof of Proposition 5.4.6.12. Let $\sigma , \sigma ': \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$ be $n$-simplices of $\operatorname{\mathcal{D}}$. We write $\sigma \sim \sigma '$ if there exists an isomorphism $\sigma \rightarrow \sigma '$ in the $\infty $-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}})$ is an identity morphism. Note that, if this condition is satisfied, then we must have $\sigma |_{ \operatorname{\partial \Delta }^ n } = \sigma ' |_{ \operatorname{\partial \Delta }^ n }$. In particular, if $\sigma $ and $\sigma '$ are both degenerate, we must have $\sigma = \sigma '$. Let $R(n)$ denote a collection of $n$-simplices of $\operatorname{\mathcal{D}}$ which contains all degenerate $n$-simplices, and contains exactly one element of every $\sim $-class. We let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ denote the simplicial subset consisting of all simplices $\tau : \Delta ^{m} \rightarrow \operatorname{\mathcal{D}}$ having the property that, for every morphism of linearly ordered sets $\alpha : [n] \rightarrow [m]$, the $n$-simplex $\Delta ^{n} \rightarrow \Delta ^{m} \xrightarrow { \tau } \operatorname{\mathcal{D}}$ belongs to $R(n)$ (by construction, it suffices to check this in the case where $\alpha $ is injective). To complete the proof, it will suffice to establish the following:

$(1)$

The simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category.

$(2)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is minimal.

$(3)$

The inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.

We begin by proving $(1)$. Suppose we are given integers $0 < i < n$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{D}}$ is an $\infty $-category, we can extend $\sigma _0$ to an $n$-simplex $\sigma '': \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$. Let $\overline{\sigma }'' = d_ i( \sigma '' )$ denote the $i$th face of $\sigma ''$. Then there is a unique element $\overline{\sigma }' \in R(n-1)$ satisfying $\overline{\sigma }' \sim \overline{\sigma }''$. Choose an isomorphism $\overline{\alpha }: \overline{\sigma }' \rightarrow \overline{\sigma }''$ in the $\infty $-category $\operatorname{Fun}( \Delta ^{n-1}, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^{n-1}, \operatorname{\mathcal{D}})$ is an identity morphism. Then $\overline{\alpha }$ can be lifted uniquely to an isomorphism $\widetilde{\alpha }: \widetilde{\sigma }' \rightarrow \sigma ''|_{ \operatorname{\partial \Delta }^{n} }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}( \Lambda ^{n}_{i}, \operatorname{\mathcal{D}})$ is an identity morphism. Applying Proposition 4.4.5.8, we can lift $\widetilde{\alpha }$ to an isomorphism $\alpha : \sigma ' \rightarrow \sigma ''$ in the $\infty $-category $\operatorname{Fun}( \Delta ^{n}, \operatorname{\mathcal{D}})$. By construction, the restriction $\sigma '|_{ \operatorname{\partial \Delta }^{n} }$ factors through $\operatorname{\mathcal{C}}$. Let $\sigma $ be the unique $n$-simplex of $\operatorname{\mathcal{D}}$ which belongs to $R(n)$ and satisfies $\sigma \sim \sigma '$. Then $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$ satisfying $\sigma |_{ \Lambda ^{n}_{i} } = \sigma '|_{ \Lambda ^{n}_{i} } = \sigma ''|_{ \Lambda ^ n_{i} } = \sigma _0$. This completes the proof of $(1)$.

We now prove $(2)$. Let $\sigma $ and $\sigma '$ be $n$-simplices of $\operatorname{\mathcal{C}}$, and suppose that there exists an isomorphism $\sigma \rightarrow \sigma '$ in $\operatorname{Fun}( \Delta ^{n}, \operatorname{\mathcal{C}})$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})$ is an identity morphism. It follows that, when regarded as $n$-simplices of $\operatorname{\mathcal{D}}$, we have $\sigma \sim \sigma '$. Since $\sigma $ and $\sigma '$ both belong to $R(n)$, we conclude that $\sigma = \sigma '$.

To prove $(3)$, we will show that $\operatorname{\mathcal{C}}$ is a deformation retract of $\operatorname{\mathcal{D}}$; that is, there exists a functor $H: \Delta ^{1} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ satisfying the following conditions:

$(i)$

The restriction $H|_{ \{ 0\} \times \operatorname{\mathcal{D}}}$ is the identity functor $\operatorname{id}_{ \operatorname{\mathcal{D}}}$.

$(ii)$

The restriction $H|_{ \{ 1\} \times \operatorname{\mathcal{D}}}$ factors through $\operatorname{\mathcal{C}}$.

$(iii)$

The restriction $H|_{ \Delta ^{1} \times \operatorname{\mathcal{C}}}$ coincides with the projection map

\[ \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}. \]

$(iv)$

For each object $D \in \operatorname{\mathcal{D}}$, the restriction $H|_{ \Delta ^1 \times \{ D\} }$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Note that these conditions guarantee that the functor $H|_{ \{ 1\} \times \operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a homotopy inverse to the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$.

Let $Q$ denote the set of pairs $(S, H_{S} )$, where $S \subseteq \operatorname{\mathcal{D}}$ is a simplicial subset which contains $\operatorname{\mathcal{C}}$ and $H_{S}: \Delta ^{1} \times S \rightarrow \operatorname{\mathcal{D}}$ is a morphism of simplicial sets which satisfies the analogues of conditions $(i)$ through $(iv)$. We regard $Q$ as a partially ordered set, where $(S, H_ S) \leq (S', H_{S'} )$ if $S \subseteq S'$ and $H_{S} = H_{S'} |_{ \Delta ^1 \times S}$. This partially ordered set satisfies the hypotheses of Zorn's lemma, and therefore contains a maximal element $(S_{\mathrm{max}}, H_{\mathrm{max}} )$. To complete the proof, it will suffice to show that $S_{\mathrm{max}} = \operatorname{\mathcal{D}}$. Assume otherwise. Then there is some $n$-simplex $\tau : \Delta ^{n} \rightarrow \operatorname{\mathcal{D}}$ which is not contained in $S_{\mathrm{max}}$. Choose $n$ as small as possible, so that $\tau _0 = \tau |_{ \operatorname{\partial \Delta }^{n} }$ factors through $S_{\mathrm{max}}$. Then the composite map

can be viewed as an isomorphism $\alpha _0: \tau _0 \rightarrow \tau '_0$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{D}})$, where $\tau '_0$ belongs to $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})$. Using Proposition 4.4.5.8, we can lift $\alpha _0$ to an isomorphism $\tau \rightarrow \tau '$ in the $\infty $-category $\operatorname{Fun}( \Delta ^{n}, \operatorname{\mathcal{D}})$. Let $\tau ''$ be the unique $n$-simplex of $\operatorname{\mathcal{D}}$ which belongs to $R(n)$ and satisfies $\tau ' \sim \tau ''$. Then there exists an isomorphism $\beta : \tau ' \rightarrow \tau ''$ in the $\infty $-category $\operatorname{Fun}( \Delta ^{n}, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{D}})$ is an identity morphism. Using Theorem 1.4.6.1, we can lift the degenerate $2$-simplex

of $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{D}})$ to a $2$-simplex

in the $\infty $-category $\operatorname{Fun}( \Delta ^{n}, \operatorname{\mathcal{D}})$. Let $S$ denote the simplicial subset of $\operatorname{\mathcal{D}}$ given by the union of $S_{\mathrm{max}}$ with the image of $\tau $. Then $H_{\mathrm{max}}$ extends uniquely to a morphism $H_{S}: \Delta ^1 \times S \rightarrow \operatorname{\mathcal{D}}$ for which the composite map

coincides with $\gamma $. By construction, the pair $(S, H_{S} )$ is an element of $Q$ satisfying $(S, H_ S) > ( S_{\mathrm{max}}, H_{ \mathrm{max}} )$, contradicting the maximality of $( S_{\mathrm{max}}, H_{ \mathrm{max}} )$. $\square$