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9.2.2 Example: Discrete and Subterminal Objects

We now consider some important special cases of Definition 9.2.1.1.

Definition 9.2.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will say that an object $X \in \operatorname{\mathcal{C}}$ is discrete if, for every object $C \in \operatorname{\mathcal{C}}$, every connected component of the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is contractible.

Definition 9.2.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will 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.

Remark 9.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then:

  • An object $X \in \operatorname{\mathcal{C}}$ is discrete (in the sense of Definition 9.2.2.1) if and only if is $0$-truncated (in the sense of Definition 9.2.2.1).

  • An object $X \in \operatorname{\mathcal{C}}$ is subterminal (in the sense of Definition 9.2.2.2) if and only if it is $(-1)$-truncated.

See Examples 3.5.7.4 and 3.5.7.5.

Example 9.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every final object of $\operatorname{\mathcal{C}}$ is subterminal, and every subterminal object of $\operatorname{\mathcal{C}}$ is discrete.

Example 9.2.2.5. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ (Construction 5.5.1.1). Then:

  • The Kan complex $X$ is a discrete object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 9.2.2.1) 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 9.2.2.2) if and only if $X$ is either empty or contractible.

See Example 9.2.1.4.

Example 9.2.2.6. Let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ be the nerve of an ordinary category $\operatorname{\mathcal{C}}_0$. Then:

  • Every object of $\operatorname{\mathcal{C}}$ is discrete.

  • An object $X \in \operatorname{\mathcal{C}}$ is subterminal (in the sense of Definition 9.2.2.2) if and only if it is subterminal in the sense of classical category theory: that is, for every object $Y \in \operatorname{\mathcal{C}}$, there is at most one morphism from $Y$ to $X$.

We now record a partial converse to Example 9.2.2.6.

Definition 9.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is locally discrete if every object $X \in \operatorname{\mathcal{C}}$ is discrete.

Note that an $\infty $-category $\operatorname{\mathcal{C}}$ is locally discrete if and only if it is locally $0$-truncated, in the sense of Definition 4.8.2.1. Invoking Corollary 4.8.2.15, we obtain the following:

Remark 9.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

  • The $\infty $-category $\operatorname{\mathcal{C}}$ is locally discrete.

  • The comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is a trivial Kan fibration.

  • There exists an ordinary category $\operatorname{\mathcal{C}}_0$ and an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$.

Example 9.2.2.9. Let $\operatorname{\mathcal{QC}}$ be the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). Then an object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}$ is discrete (in the sense of Definition 9.2.2.1) if and only if it satisfies the following pair of conditions:

  • The $\infty $-category $\operatorname{\mathcal{C}}$ is locally discrete: that is, there exists an equivalence $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$, where $\operatorname{\mathcal{C}}_0$ is an ordinary category (Remark 9.2.2.8).

  • For every object $X \in \operatorname{\mathcal{C}}_0$, the automorphism group $\operatorname{Aut}(X)$ is trivial.

See Proposition 9.2.1.17. Beware that the second condition cannot be omitted.

Remark 9.2.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X \in \operatorname{\mathcal{C}}$ be an object, and let $Y \in \operatorname{\mathcal{C}}$ be a retract of $X$. If $X$ is discrete, then $Y$ is also discrete. If $X$ is subterminal, then $Y$ is also subterminal. See Remark 9.2.1.5.

Remark 9.2.2.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories, and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:

  • If $F(X)$ is a discrete object of $\operatorname{\mathcal{D}}$, then $X$ is a discrete object of $\operatorname{\mathcal{C}}$.

  • If $F(X)$ is a subterminal object of $\operatorname{\mathcal{D}}$, then $X$ is a subterminal object of $\operatorname{\mathcal{C}}$.

In both cases, the converse holds if $F$ is an equivalence of $\infty $-categories.

Remark 9.2.2.12. 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 $m \geq 3$:

$(\ast _ m)$

Every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.

In this case, $X$ is subterminal if and only if it also satisfies condition $(\ast _2)$. See Proposition 9.2.1.16.

Remark 9.2.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An object $X \in \operatorname{\mathcal{C}}$ is subterminal if and only if the diagram

\[ X \xleftarrow { {\operatorname{id}}_ X } X \xrightarrow { {\operatorname{id}}_{X} } X \]

exhibits $X$ as a product of $X$ with itself. See Remark 9.2.1.10.

Remark 9.2.2.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

  • If $F$ preserves finite limits, then it carries discrete objects of $\operatorname{\mathcal{C}}$ to discrete objects of $\operatorname{\mathcal{D}}$ (see Remark 9.2.1.12).

  • If $F$ preserves pairwise products, then it carries subterminal objects of $\operatorname{\mathcal{C}}$ to subterminal objects of $\operatorname{\mathcal{D}}$ (see Remark 9.2.2.13).

Notation 9.2.2.15 (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 $\operatorname{\mathcal{C}}$.

Let $\mathrm{Disc}(\operatorname{\mathcal{C}})$ denote the homotopy category of $\operatorname{\mathcal{C}}^{\heartsuit }$. By construction, the $\infty $-category $\operatorname{\mathcal{C}}^{\heartsuit }$ is locally discrete, so Remark 9.2.2.8 guarantees that the comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ restricts to a trivial Kan fibration

\[ \operatorname{\mathcal{C}}^{\heartsuit } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Disc}(\operatorname{\mathcal{C}}) ). \]

For this reason, we will often abuse terminology by identifying the heart $\operatorname{\mathcal{C}}^{\heartsuit }$ with the ordinary category $\mathrm{Disc}(\operatorname{\mathcal{C}})$, which we also refer to as the heart of $\operatorname{\mathcal{C}}$.

Notation 9.2.2.16. 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 $\operatorname{Sub}(\operatorname{\mathcal{C}})$.

Remark 9.2.2.17. 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, the partially ordered set $\operatorname{Sub}(\operatorname{\mathcal{C}})$ can be identified with the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$.

Remark 9.2.2.18. 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 $X \times Y$.