Definition 10.2.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An object $X \in \operatorname{\mathcal{C}}$ is subterminal if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is either empty or contractible.
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10.2.2 Subterminal Objects
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that an object $X \in \operatorname{\mathcal{C}}$ is final if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is contractible (Definition 4.6.7.1). In this section, we study a slight variant of this condition.
Remark 10.2.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$ (Remark 10.2.1.13): that is, the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $C$ for which the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is nonempty. Then $X$ is a subterminal object of $\operatorname{\mathcal{C}}$ (in the sense of Definition 10.2.2.1) if and only if it is a final object of $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 4.6.7.1).
Example 10.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every final object of $\operatorname{\mathcal{C}}$ is subterminal.
Example 10.2.2.4. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.6.1.1). A Kan complex $X$ is subterminal (when viewed as an object of $\operatorname{\mathcal{S}}$) if and only if it is either empty or contractible.
Remark 10.2.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$ which are isomorphic. Then $X$ is subterminal if and only if $Y$ is subterminal.
Remark 10.2.2.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories, and let $X$ be an object of $\operatorname{\mathcal{C}}$. If $F(X)$ is a subterminal object of $\operatorname{\mathcal{D}}$, then $X$ is a subterminal object of $\operatorname{\mathcal{C}}$. The converse holds if $F$ is an equivalence of $\infty $-categories.
Remark 10.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is subterminal if and only if it satisfies the following condition for every integer $n \geq 2$:
Every morphism $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (n) = X$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.
To prove this, we can replace $\operatorname{\mathcal{C}}$ by the sieve $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ generated by $X$. In this case, the object $X$ satisfies condition $(\ast _ n)$ also in the case $n =1$, and is subterminal if and only if it is a final object of $\operatorname{\mathcal{C}}$ (Remark 10.2.2.2). The desired result now follows from Corollary 4.6.7.13.
Remark 10.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$ (Remark 10.2.1.13). Then $\operatorname{\mathcal{C}}^{0}$ is the essential image of the forgetful functor $U: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$. In particular, $U$ determines a functor $U^{0}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}^{0}$. In this case, the following conditions are equivalent:
The object $X \in \operatorname{\mathcal{C}}$ is subterminal (in the sense of Definition 10.2.2.1).
The functor $U^{0}$ is a trivial Kan fibration.
The functor $U^{0}$ is an equivalence of $\infty $-categories.
The equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Remark 10.2.2.7. Note that $U^0$ is a pullback of $U$, and therefore a right fibration (Proposition 4.3.6.1). In particular, $U^{0}$ is an isofibration (Example 4.4.1.10), so the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20.
Remark 10.2.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Suppose that $\operatorname{\mathcal{C}}$ admits finite limits, so that $f$ admits a Čech nerve $\operatorname{\check{C}}(X/Y)_{\bullet }$ (Proposition 10.1.4.14). If $Y$ is a subterminal object of $\operatorname{\mathcal{C}}$, then the underlying simplicial object of $\operatorname{\check{C}}(X/Y)_{\bullet }$ is also a Čech nerve of the object $X$. This follows by combining Remark 10.2.2.8 with Example 10.2.1.10.
Proposition 10.2.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An object $X \in \operatorname{\mathcal{C}}$ is subterminal if and only if the diagram exhibits $X$ as a product of $X$ with itself (in the sense of Definition 7.6.1.3).
\[ X \xleftarrow { {\operatorname{id}}_ X } X \xrightarrow { {\operatorname{id}}_{X} } X \]
Proof. Fix an object $C \in \operatorname{\mathcal{C}}$. Applying Corollary 3.2.8.7, we see that the diagonal map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \]
is a homotopy equivalence if and only if the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is either empty or contractible. Proposition 10.2.2.10 now follows by allowing the object $C$ to vary. $\square$
Corollary 10.2.2.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ preserves pairwise products, then it carries subterminal objects of $\operatorname{\mathcal{C}}$ to subterminal objects of $\operatorname{\mathcal{D}}$.
Notation 10.2.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\operatorname{Sub}(\operatorname{\mathcal{C}})$ denote the set of isomorphism classes of subterminal objects of $\operatorname{\mathcal{C}}$. If $X$ is a subterminal object of $\operatorname{\mathcal{C}}$, we let $[X] \in \operatorname{Sub}(\operatorname{\mathcal{C}})$ denote its isomorphism class. Given a pair of subterminal objects $X$ and $X'$, we write $[X] \subseteq [X']$ if there exists a morphism $f: X \rightarrow X'$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Note that the relation $\subseteq $ is a partial ordering on the set $\operatorname{Sub}(\operatorname{\mathcal{C}})$.
Remark 10.2.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category.
If $\operatorname{\mathcal{C}}$ has a final object, then the partially ordered set $\operatorname{Sub}(\operatorname{\mathcal{C}})$ has a largest element: namely, the isomorphism class $[X]$, where $X$ is any final object of $\operatorname{\mathcal{C}}$.
If $\operatorname{\mathcal{C}}$ admits finite products, then $\operatorname{Sub}(\operatorname{\mathcal{C}})$ is a lower semilattice: that is, every finite subset of $\operatorname{Sub}(\operatorname{\mathcal{C}})$ has a greatest lower bound. In particular, every pair of elements $[X], [Y] \in \operatorname{Sub}(\operatorname{\mathcal{C}})$ have a greatest lower bound which we will denote by $[X] \cap [Y]$, given by the isomorphism class of the product $X \times Y$.
Proposition 10.2.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the subterminal objects of $\operatorname{\mathcal{C}}$. Then the construction $X \mapsto [X]$ induces a trivial Kan fibration $U: \operatorname{\mathcal{C}}' \rightarrow \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$.
Proof. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}'$. Since $Y$ is subterminal, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is either empty or contractible. It follows that the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )}( [X], [Y] )$ is a homotopy equivalence. Allowing $X$ and $Y$ to vary, we deduce that the functor $U$ is fully faithful. By construction, $U$ is surjective on objects, and therefore essentially surjective. Applying Theorem 4.6.2.19, we conclude that $U$ is an equivalence of $\infty $-categories. Since $\operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ is the nerve of a category, $U$ is automatically an inner fibration (Proposition 4.1.1.10). Moreover, every isomorphism in $\operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ is an identity morphism, so $U$ is an isofibration. Applying Proposition 4.5.5.20, we conclude that $U$ is a trivial Kan fibration. $\square$
Corollary 10.2.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
There exists a partially ordered set $A$ and an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(A)$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is either empty or contractible.
Every object of $\operatorname{\mathcal{C}}$ is subterminal.
Proof. The implications $(1) \Rightarrow (2)$ and $(2) \Leftrightarrow (3)$ follow immediately from the definitions. We conclude by observing that if condition $(3)$ is satisfied, then the construction $X \mapsto [X]$ induces a trivial Kan fibration $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ (Proposition 10.2.2.14). $\square$