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4.6.6 Initial and Final Objects

Let $\operatorname{\mathcal{C}}$ be a category. Recall that an object $Y \in \operatorname{\mathcal{C}}$ is initial if, for every object $Z \in \operatorname{\mathcal{C}}$, there is a unique morphism from $Y$ to $Z$. This definition has an obvious counterpart in the setting of $\infty $-categories.

Definition 4.6.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that an object $Y \in \operatorname{\mathcal{C}}$ is initial if, for every object $Z \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is a contractible Kan complex. We say that $Y$ is final if, for every object $X \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a contractible Kan complex.

Remark 4.6.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if it is final when viewed as an object of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Example 4.6.6.3. Let $\operatorname{\mathcal{C}}$ be a category. An object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if it is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Similarly, an object $Y \in \operatorname{\mathcal{C}}$ is final if and only if it is final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Example 4.6.6.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denote their join (Construction 4.3.3.13). Then $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is also an $\infty $-category (Corollary 4.3.3.24). It follows from Example 4.6.1.5 that if $X$ is an initial object of $\operatorname{\mathcal{C}}$, then it is also initial when regarded as an object of $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$. Similarly, if $Y$ is a final object of $\operatorname{\mathcal{D}}$, then it is also final when regarded as an object of $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$.

Example 4.6.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the cone point of the $\infty $-category $\operatorname{\mathcal{C}}^{\triangleleft }$ is an initial object. Similarly, the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ is a final object.

Remark 4.6.6.6. In the formulation of Definition 4.6.6.1, we can replace the Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ by their left-pinched variants $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(Y,Z)$, or by their right-pinched variants $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(Y,Z)$ (see Proposition 4.6.5.9).

Example 4.6.6.7. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, so that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1). Combining Remark 4.6.6.6 with Theorem 4.6.7.5, we deduce the following:

  • An object $Y \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$ is contractible.

  • An object $Y \in \operatorname{\mathcal{C}}$ final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is contractible.

Example 4.6.6.8. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (Construction 5.6.1.1). Then:

  • The Kan complex $X$ is an initial object of the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if it is empty.

  • The Kan complex $X$ is a final object of the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if it is contractible.

Example 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1). Combining Remark 4.6.6.6) with Example 4.6.5.12, we obtain the following:

  • An object $Y \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible (that is, there exists a $1$-morphism from $Y$ to $Z$ and for every pair of morphisms $f,g: X \rightarrow Y$, there is a unique isomorphism $\gamma : f \xRightarrow {\sim } g$).

  • An object $Y \in \operatorname{\mathcal{C}}$ is final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible.

Proposition 4.6.6.10. Let $\operatorname{\mathcal{C}}$ be a differential graded category, so that the differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.5.3.10). Let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $Y$ is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

$(2)$

The object $Y$ is final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

$(3)$

The identity morphism $\operatorname{id}_{Y}: Y \rightarrow Y$ is nullhomologous: that is, there exists a $1$-chain $e \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{1}$ satisfying $\partial (e) = \operatorname{id}_{Y}$.

Proof. We will show that $(1) \Leftrightarrow (3)$; the proof that $(2) \Leftrightarrow (3)$ is similar. If condition $(1)$ is satisfied, then there exists a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ with boundary as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{0} & \\ Y \ar [ur]^{0} \ar [rr]^{ \operatorname{id}_ Y } & & Y, } \]

which we can identify with a $1$-chain $e \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{1}$ satisfying $\partial (e) = \operatorname{id}_ Y$ (see Example 2.5.3.4). Conversely, suppose that there exists $e \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{1}$ satisfying $\partial (e) = \operatorname{id}_{Y}$. For every object $Z \in \operatorname{\mathcal{C}}$, $e$ determines a chain homotopy from the identity map $\operatorname{id}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast }$ to the zero map. It follows that the homology of chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast }$ vanishes, so that the Eilenberg-MacLane space $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } )$ of Construction 2.5.6.3 is a contractible Kan complex. Example 4.6.5.14 supplies an isomorphism of Kan complexes $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}}(Y,Z) \simeq \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } )$. Allowing $Z$ to vary and invoking Remark 4.6.6.6, we conclude that $Y$ is an initial object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. $\square$

Proposition 4.6.6.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The object $Y$ is initial if and only if the projection map $\operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets.

$(2)$

The object $Y$ is final if and only if the projection map $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets.

Proof. We will give the proof of $(1)$; the proof of $(2)$ is similar. Proposition 4.3.6.1 guarantees that the projection map $q: \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets. Applying Proposition 4.4.2.14, we see that $q$ is a trivial Kan fibration if and only if, for each object $Z \in \operatorname{\mathcal{C}}$, the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( Y, Z) = \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $ is a contractible Kan complex. By virtue of Remark 4.6.6.6, this is equivalent to the assumption that $Y$ is an initial object of $\operatorname{\mathcal{C}}$. $\square$

Corollary 4.6.6.12. Let $X$ be a Kan complex and let $x \in X$ be a vertex. The following conditions are equivalent:

$(1)$

The vertex $x$ is initial when viewed as an object of the $\infty $-category $X$.

$(2)$

The vertex $x$ is final when viewed as an object of the $\infty $-category $X$.

$(3)$

The Kan complex $X$ is contractible.

In particular, these conditions are independent of the choice of vertex $x \in X$.

Proof. If the Kan complex $X$ is contractible, then the projection map $X_{x/} \rightarrow X$ is a trivial Kan fibration (Corollary 4.3.7.19), so the object $x \in X$ is initial by virtue of Proposition 4.6.6.11. Conversely, if the projection map $X_{x/} \rightarrow X$ is a trivial Kan fibration, then it is a homotopy equivalence (Proposition 3.1.6.10). Since the Kan complex $X_{x/}$ is contractible (Corollary 4.3.7.14), it follows that $X$ is contractible. This proves the equivalence of $(1)$ and $(3)$; the equivalence of $(2)$ and $(3)$ follows by a similar argument. $\square$

Corollary 4.6.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, let $U: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let $Y$ be an initial object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

There exists an object $\widetilde{Y} \in \operatorname{\mathcal{C}}_{/f}$ satisfying $U( \widetilde{Y} ) = Y$.

$(2)$

If $\widetilde{Y}$ is any object of $\operatorname{\mathcal{C}}_{/f}$ satisfying $U( \widetilde{Y} ) = Y$, then $\widetilde{Y}$ is an initial object of $\operatorname{\mathcal{C}}_{/f}$.

Proof. Assertion $(1)$ is equivalent to the statement that $f$ can be lifted to a map $\widetilde{f}: K \rightarrow \operatorname{\mathcal{C}}_{Y/}$. This is clear, since the projection map $\operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (Proposition 4.6.6.11). To prove $(2)$, fix an object $\widetilde{Y} \in \operatorname{\mathcal{C}}_{ / f}$ satisfying $U( \widetilde{Y} ) = Y$. By virtue of Proposition 4.6.6.11, it will suffice to show that the projection map $(\operatorname{\mathcal{C}}_{/f})_{\widetilde{Y}/ } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a trivial Kan fibration. Equivalently, we wish to show that every lifting problem

4.58
\begin{equation} \begin{gathered}\label{equation:coslices-create-final} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & (\operatorname{\mathcal{C}}_{/f})_{\widetilde{Y}/ } \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}_{/f} } \end{gathered} \end{equation}

admits a solution, provided that the left vertical map is a monomorphism. Unwinding the definitions, we can rewrite (4.58) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \star K \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{Y/} \ar [d] \\ B \star K \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}. } \]

Our assumption that the object $Y \in \operatorname{\mathcal{C}}$ is initial guarantees that this lifting problem has a solution (Proposition 4.6.6.11). $\square$

Corollary 4.6.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if, for every integer $n \geq 1$ and every morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (0) = Y$, there exists an $n$-simplex $\overline{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\sigma }|_{ \operatorname{\partial \Delta }^ n} = \sigma $.

Proof. Let $n$ be a positive integer. Using the isomorphism

\[ \operatorname{\partial \Delta }^{n} \simeq (\emptyset \star \Delta ^{n-1} ) \coprod _{ (\emptyset \star \operatorname{\partial \Delta }^{n-1}) } ( \Delta ^{0} \star \operatorname{\partial \Delta }^{n-1} ) \]

supplied by Variant 4.3.6.16, we see that a morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (0) = Y$ can be identified with a commutative diagram

4.59
\begin{equation} \begin{gathered}\label{equation:characterize-initial-by-slice} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n-1} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{Y/} \ar [d] \\ \Delta ^{n-1} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

and that an extension of $\sigma $ to an $n$-simplex of $\operatorname{\mathcal{C}}$ can be identified with a dotted arrow which renders the diagram commutative. By virtue of Proposition 4.6.6.11, the object $Y$ is initial if and only if every lifting problem of the form (4.59) admits a solution: that is, if and only if the projection map $\operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets. $\square$

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which contains an initial object $X$. This object is rarely unique: every object $Y \in \operatorname{\mathcal{C}}$ which is isomorphic to $X$ is also initial (Corollary 4.6.6.16). However, the object $X$ is essentially unique in the following sense:

Corollary 4.6.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the initial objects of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{\mathrm{fin}} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the final objects of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ contains an initial object, then $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex. If $\operatorname{\mathcal{C}}$ contains a final object, then $\operatorname{\mathcal{C}}^{\mathrm{fin}}$ is a contractible Kan complex.

Proof. Assume that $\operatorname{\mathcal{C}}$ contains an initial object, we will show that $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex (the analogous assertion for final objects follows by a similar argument). Suppose we are given a morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}^{\mathrm{init} }$; we wish to show that $\sigma $ can be extended to a morphism $\overline{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}^{\mathrm{init}}$. If $n=0$, this follows from our assumption that $\operatorname{\mathcal{C}}$ contains an initial object. If $n > 0$, then we can regard $\sigma $ as a morphism from $\operatorname{\partial \Delta }^ n$ to $\operatorname{\mathcal{C}}$ with the property that $\sigma (i) \in \operatorname{\mathcal{C}}$ is initial for $0 \leq i \leq n$. Setting $i=0$, we conclude that $\sigma $ can be extended to a morphism $\overline{\sigma }: \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$, which automatically factors through the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$. $\square$

Corollary 4.6.6.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an initial object of $\operatorname{\mathcal{C}}$. Then an object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if it is isomorphic to $X$.

Proof. If $X$ and $Y$ are initial objects of $\operatorname{\mathcal{C}}$, then they are contained in the contractible Kan complex $\operatorname{\mathcal{C}}^{\mathrm{init}}$ of Corollary 4.6.6.15 and are therefore isomorphic when viewed as objects of $\operatorname{\mathcal{C}}$. Conversely, suppose that $X$ is initial and that there exists an isomorphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$; we wish to show that $Y$ is also initial. Fix an object $Z \in \operatorname{\mathcal{C}}$; we wish to show that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible. Let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (see Construction 4.6.8.13). Since $f$ an isomorphism in $\operatorname{\mathcal{C}}$, its homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, so composition with $[f]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the category $\mathrm{h} \mathit{\operatorname{Kan}}$. Since the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is contractible, it follows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is also contractible. $\square$

Notation 4.6.6.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often use the symbol $\emptyset _{\operatorname{\mathcal{C}}}$ to denote an initial object of $\operatorname{\mathcal{C}}$, provided that such an object exists. In this case, we will sometimes abuse terminology by referring to $\emptyset _{\operatorname{\mathcal{C}}}$ as the initial object of $\operatorname{\mathcal{C}}$. This abuse is justified by Corollary 4.6.6.15, which guarantees that $\emptyset _{\operatorname{\mathcal{C}}}$ is uniquely determined up to a contractible space of choices (in particular, it is well-defined up to isomorphism). Similarly, we will often use the symbol ${\bf 1}_{\operatorname{\mathcal{C}}}$ to denote a final object of $\operatorname{\mathcal{C}}$, provided that such an object exists, and will sometimes abuse terminology by referring to ${\bf 1}_{\operatorname{\mathcal{C}}}$ as the final object of $\operatorname{\mathcal{C}}$. When it is unlikely to cause confusion, we will sometimes omit the subscripts and denote the objects $\emptyset _{\operatorname{\mathcal{C}}}$ and ${\bf 1}_{\operatorname{\mathcal{C}}}$ by $\emptyset $ and ${\bf 1}$, respectively.

Corollary 4.6.6.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

If $X$ is initial as an object of the $\infty $-category $\operatorname{\mathcal{C}}$, then it is also initial when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

$(2)$

If $\operatorname{\mathcal{C}}$ has an initial object and $X$ is initial as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, then $X$ is initial as an object of the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. Assertion $(1)$ is immediate from the definition. To prove $(2)$, assume that $\operatorname{\mathcal{C}}$ has an initial object $Y$. Then $Y$ is also initial when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. If $X$ is an initial object of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, then $X$ and $Y$ are isomorphic when viewed as objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, hence also when viewed as objects of the $\infty $-category $\operatorname{\mathcal{C}}$. Invoking Corollary 4.6.6.16, we conclude that $X$ is also an initial object of $\operatorname{\mathcal{C}}$. $\square$

Warning 4.6.6.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$ which is initial as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Then $X$ need not be initial when viewed as an object of $\operatorname{\mathcal{C}}$. For example, if $\operatorname{\mathcal{C}}$ is simply connected Kan complex, then every object $X \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} = \pi _{\leq 1}(\operatorname{\mathcal{C}})$. However, $X$ is initial as an object of $\operatorname{\mathcal{C}}$ only if $\operatorname{\mathcal{C}}$ is contractible (Corollary 4.6.6.12).

Proposition 4.6.6.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

If $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$, then $Y$ is an initial object of $\operatorname{\mathcal{C}}$.

$(2)$

If $F(Y)$ is a final object of $\operatorname{\mathcal{D}}$, then $Y$ is a final object of $\operatorname{\mathcal{C}}$.

Proof. Let $Z$ be an object of $\operatorname{\mathcal{C}}$. If $F(Y)$ is an initial object of the $\infty $-category $\operatorname{\mathcal{D}}$, then the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) )$ is a contractible Kan complex. Since $F$ is fully faithful, it follows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is also a contractible Kan complex. Allowing $Z$ to vary, we conclude that $Y$ is an initial object of $\operatorname{\mathcal{C}}$. This proves $(1)$; the proof of $(2)$ is similar. $\square$

Corollary 4.6.6.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The object $Y$ is initial if and only if $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$.

$(2)$

The object $Y$ is final if and only if $F(Y)$ is a final object of $\operatorname{\mathcal{D}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Note that since $F$ is an equivalence of $\infty $-categories, it is fully faithful (Theorem 4.6.2.17). If $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$, then Proposition 4.6.6.20 guarantees that the object $Y \in \operatorname{\mathcal{C}}$ is initial. To prove the converse, let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse of the functor $F$. Then $G \circ F$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$, so that $(G \circ F)(Y)$ is isomorphic to $Y$ as an object of the $\infty $-category $\operatorname{\mathcal{C}}$. If $Y$ is an initial object of $\operatorname{\mathcal{C}}$, then Corollary 4.6.6.16 guarantees that $(G \circ F)(Y)$ is also an initial object of $\operatorname{\mathcal{C}}$. Since the equivalence $G$ is fully faithful (Theorem 4.6.2.17), Proposition 4.6.6.20 guarantees that $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$. $\square$

Corollary 4.6.6.22. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are equivalent. Then:

  • The $\infty $-category $\operatorname{\mathcal{C}}$ has an initial object if and only if the $\infty $-category $\operatorname{\mathcal{D}}$ has an initial object.

  • The $\infty $-category $\operatorname{\mathcal{C}}$ has a final object if and only if the $\infty $-category $\operatorname{\mathcal{D}}$ has a final object.

Proposition 4.6.6.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Definition 1.3.6.1).

$(2)$

The morphism $f$ is final when regarded as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$.

$(2')$

The morphism $f$ is final when regarded as an object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $.

$(3)$

The morphism $f$ is initial when regarded as an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$.

$(3')$

The morphism $f$ is initial when regarded as an object of the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$.

Proof. The equivalences $(2) \Leftrightarrow (2')$ and $(3) \Leftrightarrow (3')$ follow from Corollaries 4.6.4.18 and 4.6.6.21. We will complete the proof by showing that $(1) \Leftrightarrow (3)$; the equivalence $(1) \Leftrightarrow (2)$ follows by applying the same argument in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. By virtue of Corollary 4.6.6.14, condition $(3)$ is equivalent to the requirement that the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{X/}$ is a trivial Kan fibration: that is, every lifting problem

4.60
\begin{equation} \begin{gathered}\label{equation:limit-of-point} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{f/} \ar [d] \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}_{X/} } \end{gathered} \end{equation}

admits a solution. Using the isomorphism of simplicial sets

\[ (\Delta ^1 \star \operatorname{\partial \Delta }^ n) \coprod _{ \{ 0\} \star \operatorname{\partial \Delta }^ n } ( \{ 0\} \star \Delta ^ n ) \simeq \Lambda ^{n+2}_{0} \]

supplied by Lemma 4.3.6.14, we can identify (4.60) with a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+2}_{0} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^{n+2} \ar@ {-->}[ur]^{\sigma } \ar [r] & \Delta ^0, } \]

where $\sigma _0$ carries the initial edge $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{n+2}_{0}$ to the morphism $f$. The equivalence $(1) \Leftrightarrow (3)$ now follows from the criterion of Theorem 4.4.2.6. $\square$

Corollary 4.6.6.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The object $Y$ is final if and only if the projection map $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ admits a section $G$ satisfying $G(Y) = \operatorname{id}_{Y}$.

$(2)$

The object $Y$ is initial if and only if the projection map $F': \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ admits a section $G'$ satisfying $G'(Y) = \operatorname{id}_{Y}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. If $Y$ is a final object, then the projection map $F: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (Proposition 4.6.6.11), so the construction $Y \mapsto \operatorname{id}_ Y$ can be extended to a section of $F$. Conversely, suppose that $F$ admits a section $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$ satisfying $G(Y) = \operatorname{id}_ Y$. Let $X$ be an object of $\operatorname{\mathcal{C}}$: we wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible. The functors $G$ and $F$ induce morphisms of Kan complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Y}}( G(X), \operatorname{id}_ Y ) \xrightarrow {F} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y), \]

whose composition is the identity. In particular, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a retract of $ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Y}}( G(X), \operatorname{id}_ Y )$. It will therefore suffice to show that the Kan complex $ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Y}}( G(X), \operatorname{id}_ Y )$ is contractible. This is clear, since $\operatorname{id}_{Y}$ is a final object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ (Proposition 4.6.6.23). $\square$

Corollary 4.6.6.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $Y \in \operatorname{\mathcal{C}}$ is final.

$(2)$

There exists a functor $F: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $F|_{\operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and for which the composition

\[ \Delta ^1 \simeq \{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}} \]

is the identity morphism $\operatorname{id}_{Y}$ (in particular, $F$ carries the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ to the object $Y$).

$(3)$

The inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne.

Proof. The equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Corollary 4.6.6.24. We next show that $(2)$ implies $(3)$. If condition $(2)$ is satisfied, then we have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ Y\} \ar [r] \ar [d] & \{ Y\} ^{\triangleright } \ar [r] \ar [d] & \{ Y\} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } \ar [r]^-{F} & \operatorname{\mathcal{C}}} \]

where the horizontal compositions are the identity. Since the inclusion $\{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is right anodyne (Lemma 4.3.7.8), it follows that the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is also right anodyne.

We now complete the proof by showing that $(3)$ implies $(2)$. Suppose that the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne; we wish to show that there exists a functor $F: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $F|_{ \operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and $F|_{ \{ Y\} ^{\triangleright } } = \operatorname{id}_ Y$. For this, it will suffice to show that the inclusion map

\[ \operatorname{\mathcal{C}}\coprod _{ \{ Y\} } \{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \]

is inner anodyne, which is a special case of Proposition 4.3.6.4. $\square$

Corollary 4.6.6.26. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which has either an initial object or a final object. Then $\operatorname{\mathcal{C}}$ is weakly contractible.

Proof. Let $Y$ be an final object of $\operatorname{\mathcal{C}}$. Then the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne (Corollary 4.6.6.25), and therefore a weak homotopy equivalence. $\square$