Definition 4.7.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that 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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4.7.4 Discrete and Subterminal Objects
Let $\operatorname{\mathcal{C}}$ be a category. Recall that an object $X \in \operatorname{\mathcal{C}}$ is subterminal if, for every object $C \in \operatorname{\mathcal{C}}$, there is at most one morphism from $C$ to $X$. This condition has a counterpart in the setting of $\infty $-categories:
Remark 4.7.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is subterminal (in the sense of Definition 4.7.4.1) if and only if it is $(-1)$-truncated (in the sense of Definition 4.7.1.1).
Example 4.7.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every final object of $\operatorname{\mathcal{C}}$ is subterminal.
Example 4.7.4.4. Let $\operatorname{\mathcal{C}}$ be a category. Then an object $X \in \operatorname{\mathcal{C}}$ is subterminal (in the usual category-theoretic sense) if and only it is subterminal when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Remark 4.7.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ and $Y$ be isomorphic objects of $\operatorname{\mathcal{C}}$. Then $X$ is subterminal if and only if $Y$ is subterminal. See Remark 4.7.1.13.
Remark 4.7.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is subterminal if and only if $F(X)$ is a subterminal object of $\operatorname{\mathcal{D}}$. See Remark 4.7.1.14.
Remark 4.7.4.7. 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 $m \geq 2$:
Every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.
See Proposition 4.7.1.16.
Proposition 4.7.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The object $X$ is subterminal.
Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $C$ for which there exists a morphism $f: C \rightarrow X$ in $\operatorname{\mathcal{C}}$. Then $X$ is a final object of $\operatorname{\mathcal{C}}'$.
The projection map $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}'$ is a trivial Kan fibration.
The projection map $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}'$ is an equivalence of $\infty $-categories
Proof. The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definitions, the equivalence $(2) \Leftrightarrow (3)$ is a special case of Corollary 4.7.3.12, the equivalence $(3) \Leftrightarrow (4)$ follows from Proposition 4.5.6.20. $\square$
Notation 4.7.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\operatorname{Sub}(\operatorname{\mathcal{C}})$ denote the collection 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 $\operatorname{Sub}(\operatorname{\mathcal{C}})$.
Example 4.7.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}$ has a final object ${\bf 1}_{\operatorname{\mathcal{C}}}$, then the isomorphism class $[ {\bf 1}_{\operatorname{\mathcal{C}}} ]$ is a largest element of the partially ordered set $\operatorname{Sub}(\operatorname{\mathcal{C}})$.
Proposition 4.7.4.11. 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 of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$.
Stated more informally, Proposition 4.7.4.11 asserts that the full subcategory of subterminal objects of $\operatorname{\mathcal{C}}$ can be identified with the partially ordered set $\operatorname{Sub}(\operatorname{\mathcal{C}})$.
Proof of Proposition 4.7.4.11. Fix an integer $m \geq 0$; we wish to show that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}' \ar [d] \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) ) } \]
admits a solution. For $m = 0$, this follows from the definition of $\operatorname{\mathcal{C}}'$. For $m = 1$, it follows from the definition of the partial ordering on $\operatorname{Sub}(\operatorname{\mathcal{C}})$. For $m \geq 2$, it follows from the lifting property of Remark 4.7.4.7 (together with Example 3.5.3.3). $\square$
Corollary 4.7.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
Every object of $\operatorname{\mathcal{C}}$ is subterminal.
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.
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to (the nerve of) a partially ordered set.
We now study another important special case of Definition 4.7.1.1.
Definition 4.7.4.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that an object $X \in \operatorname{\mathcal{C}}$ is discrete if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is a disjoint union of contractible Kan complexes.
Remark 4.7.4.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is discrete (in the sense of Definition 4.7.4.13) if and only if is $0$-truncated (in the sense of Definition 4.7.1.1).
Example 4.7.4.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every subterminal object of $\operatorname{\mathcal{C}}$ is discrete. In particular, every final object of $\operatorname{\mathcal{C}}$ is discrete.
Example 4.7.4.16. Let $\operatorname{\mathcal{C}}$ be a category. Then every object of $\operatorname{\mathcal{C}}$ is discrete when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. See Proposition 4.7.4.22 for a partial converse.
Example 4.7.4.17. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ (Construction 3.1.6.1). Then:
The Kan complex $X$ is a discrete object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 4.7.4.13) if and only if every connected component of $X$ is contractible: that is, the projection map $X \rightarrow \pi _0(X)$ is a homotopy equivalence.
The Kan complex $X$ is a subterminal object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 4.7.4.1) if and only if $X$ is either empty or contractible.
See Example 4.7.1.8.
Remark 4.7.4.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ and $Y$ be isomorphic objects of $\operatorname{\mathcal{C}}$. Then $X$ is discrete if and only if $Y$ is discrete. See Remark 4.7.1.13.
Remark 4.7.4.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is discrete if and only if $F(X)$ is a discrete object of $\operatorname{\mathcal{D}}$. See Remark 4.7.1.14.
Remark 4.7.4.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is discrete if and only if it satisfies the following condition for every integer $m \geq 3$:
Every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.
See Proposition 4.7.1.16.
Notation 4.7.4.21 (The Heart of an $\infty $-Category). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\operatorname{\mathcal{C}}^{\heartsuit }$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the discrete objects of $\operatorname{\mathcal{C}}$. We will refer to $\operatorname{\mathcal{C}}^{\heartsuit }$ as the heart of the $\infty $-category $\operatorname{\mathcal{C}}$.
Proposition 4.7.4.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0$ denote the homotopy category of the heart $\operatorname{\mathcal{C}}^{\heartsuit }$. Then the canonical map $U: \operatorname{\mathcal{C}}^{\heartsuit } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ is a trivial Kan fibration.
Proof. Fix an integer $m \geq 0$; we wish to show that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}^{\heartsuit } \ar [d] \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 ). } \]
For $m \leq 1$, this is immediate (since the right vertical map is bijective on vertices and surjective on edges). For $m = 2$, it follows from the definition of the composition law on the homotopy category $\operatorname{\mathcal{C}}_0$ (see Notation 1.4.4.3). For $m \geq 3$, it follows from the characterization of discrete objects given in Remark 4.7.4.20. $\square$
Corollary 4.7.4.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to (the nerve of) an ordinary category.
Every object of $\operatorname{\mathcal{C}}$ is discrete: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a disjoint union of contractible Kan complexes.
The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} )$ is a trivial Kan fibration.
The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} )$ is an equivalence of $\infty $-categories.
Proof. The implication $(1) \Rightarrow (2)$ follows from Example 4.6.1.4, the implication $(2) \Rightarrow (3)$ from Proposition 4.7.4.22, the implication $(3) \Rightarrow (4)$ from Proposition 4.5.4.11, and the implication $(4) \Rightarrow (1)$ is trivial. $\square$