Definition 9.4.2.1. 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.
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9.4.2 Example: Discrete and Subterminal Objects
We now consider some important special cases of Definition 9.4.1.1.
Definition 9.4.2.2. 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 9.4.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then:
See Examples 3.5.7.4 and 3.5.7.5.
Example 9.4.2.4. 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 9.4.2.5. 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 9.4.2.1) 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 9.4.2.2) if and only if $X$ is either empty or contractible.
See Example 9.4.1.4.
Example 9.4.2.6. 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 9.4.2.2) 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 $X$.
Remark 9.4.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. If $X$ is discrete, then the canonical map $\mathrm{h} \mathit{(\operatorname{\mathcal{C}}_{/X} )} \rightarrow (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$ is an equivalence of categories. This is a special case of Corollary 9.4.1.15.
Proposition 9.4.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let be a pullback diagram in $\operatorname{\mathcal{C}}$. If the object $X$ is discrete, then (9.28) determines a pullback diagram in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
9.28
\begin{equation} \begin{gathered}\label{equation:pullback-over-discrete} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \end{gathered} \end{equation}
Beware that the conclusion of Proposition 9.4.2.8 is generally false if we do not assume that the object $X$ is discrete (see Warning 7.6.2.3).
Proof of Proposition 9.4.2.8. By virtue of Proposition 7.6.2.14, it will suffice to show that the the tautological map $F: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$ preserves products. Since $X$ is discrete, we can use Remark 9.4.2.7 to identify $F$ with the canonical map from $\operatorname{\mathcal{C}}_{/X}$ to (the nerve of) its homotopy category, which always preserves products (see Warning 7.6.1.12). $\square$
We now record a partial converse to Example 9.4.2.6.
Definition 9.4.2.9. 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 4.8.2.16, we obtain the following:
Remark 9.4.2.10. 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 9.4.2.11. 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 9.4.2.1) 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 9.4.2.10).
For every object $X \in \operatorname{\mathcal{C}}_0$, the automorphism group $\operatorname{Aut}(X)$ is trivial.
See Proposition 9.4.1.23. Beware that the second condition cannot be omitted.
Remark 9.4.2.12. 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 9.4.1.5.
Remark 9.4.2.13. 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 9.4.2.14. 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 $\operatorname{\mathcal{C}}$.
In this case, $X$ is subterminal if and only if it also satisfies condition $(\ast _2)$. See Proposition 9.4.1.18.
Remark 9.4.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a discrete object of $\operatorname{\mathcal{C}}$, and suppose we are given a diagram $F: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/X}$. If the simplicial set $K$ is connected, then $F$ is a limit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a limit diagram. In particular, the forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed colimits. If the object $X$ is subterminal, then these conclusions hold more generally if when the simplicial set. See Corollary 9.4.1.22.
Remark 9.4.2.16. 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. See Remark 9.4.1.10.
\[ X \xleftarrow { {\operatorname{id}}_ X } X \xrightarrow { {\operatorname{id}}_{X} } X \]
Remark 9.4.2.17. 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 9.4.1.12).
If $F$ preserves pairwise products, then it carries subterminal objects of $\operatorname{\mathcal{C}}$ to subterminal objects of $\operatorname{\mathcal{D}}$ (see Remark 9.4.2.16).
Notation 9.4.2.18 (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 9.4.2.10 guarantees that the comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ restricts to a trivial Kan fibration 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 $\operatorname{\mathcal{C}}$.
\[ \operatorname{\mathcal{C}}^{\heartsuit } \rightarrow \operatorname{N}_{\bullet }( \mathrm{Disc}(\operatorname{\mathcal{C}}) ). \]
Notation 9.4.2.19. 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 9.4.2.20. 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 9.4.2.21. 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 $X \times Y$.