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4.4.3 The Core of an $\infty $-Category

Let $\operatorname{\mathcal{C}}$ be a category. Recall that the core of $\operatorname{\mathcal{C}}$ is the subcategory $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$ comprised of all objects of $\operatorname{\mathcal{C}}$ and all isomorphisms between them (Construction 1.2.4.4). In this section, we generalize this construction to the setting of $\infty $-categories.

Construction 4.4.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\operatorname{\mathcal{C}}^{\simeq }$ denote the simplicial subset of $\operatorname{\mathcal{C}}$ comprised of those simplices $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ which carry each edge of $\Delta ^ n$ to an isomorphism in $\operatorname{\mathcal{C}}$. We will refer to $\operatorname{\mathcal{C}}^{\simeq }$ as the core of $\operatorname{\mathcal{C}}$.

Remark 4.4.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category, and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\simeq }$ denote the core of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Then the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$ fits into a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{\simeq } \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\simeq }) \ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ). } \]

Example 4.4.3.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, and let $\operatorname{\mathcal{C}}^{\simeq }$ denote its core (in the sense of Construction 1.2.4.4). Then the core of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Construction 4.4.3.1) can be identified with the nerve of $\operatorname{\mathcal{C}}^{\simeq }$. That is, we have a canonical isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\simeq } \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\simeq } )$.

Remark 4.4.3.4 (Functoriality). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ carries the core $\operatorname{\mathcal{C}}^{\simeq }$ into the core $\operatorname{\mathcal{D}}^{\simeq }$ (see Remark 1.4.1.6), and therefore restricts to a morphism of Kan complexes $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$.

Proposition 4.4.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is a subcategory of $\operatorname{\mathcal{C}}$ (Definition 4.1.2.2): that is, the inclusion $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration.

Proposition 4.4.3.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories. Then the induced map $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a Kan fibration.

Proof. Fix integers $n > 0$ and $0 \leq i \leq n$; we wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}^{\simeq } \ar [d]^{F^{\simeq }} \\ \Delta ^ n \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{D}}^{\simeq } } \]

admits a solution. In the case $n=1$, this follows either from our definition of isofibration (in the case $i=1$) or from Corollary 4.4.1.8 (in the case $i=0$). We may therefore assume that $n \geq 2$. We claim that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $F(\sigma ) = \overline{\sigma }$. If $0 < i < n$, this follows from the fact that $F$ is an inner fibration. The extremal cases $i=0$ and $i=n$ follow from Proposition 4.4.2.10 (applied to the inner fibration $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and its opposite $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$). To complete the proof, it will suffice to show that $\sigma $ carries each edge of $\Delta ^ n$ to an isomorphism in $\operatorname{\mathcal{C}}$. For $n > 2$, this is automatic (since the horn $\Lambda ^{n}_{i}$ contains every edge of $\Delta ^ n$). In the case $n=2$ it follows from the two-out-of-three property for isomorphisms in $\operatorname{\mathcal{C}}$ (Remark 1.3.6.3). $\square$

Corollary 4.4.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is a Kan complex.

Proof. Apply Proposition 4.4.3.6 to the inner fibration $\operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$. $\square$

Corollary 4.4.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $u$ is an isomorphism.

$(2)$

There exists a Kan complex $\operatorname{\mathcal{E}}$, a morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{E}}$, and a functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $F(\overline{u}) = u$.

$(3)$

There exists a contractible Kan complex $\operatorname{\mathcal{E}}$, a morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{E}}$, and a functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $F(\overline{u}) = u$.

Proof. If $u$ is an isomorphism, then it belongs to the image of the inclusion functor $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$. Since the core $\operatorname{\mathcal{C}}^{\simeq }$ is a Kan complex, this proves that $(1) \Rightarrow (2)$. Conversely, if we can write $u = F( \overline{u} )$ for some functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ where $\operatorname{\mathcal{E}}$ is a Kan complex, then Remark 1.4.1.6 guarantees that $u$ is an isomorphism in $\operatorname{\mathcal{C}}$ (since $\overline{u}$ is automatically an isomorphism in $\operatorname{\mathcal{E}}$, by virtue of Proposition 1.3.6.11). This proves that $(2) \Rightarrow (1)$.

The implication $(3) \Rightarrow (2)$ is immediate. We will complete the proof by showing that $(2)$ implies $(3)$. Let $\operatorname{\mathcal{E}}$ be a Kan complex, let $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor, and let $\overline{u}$ be an edge of $\operatorname{\mathcal{E}}$ satisfying $F( \overline{u} ) = u$. Let us identify $\overline{u}$ with a morphism of simplicial sets $\Delta ^1 \rightarrow \operatorname{\mathcal{E}}$. By virtue of Proposition 3.1.6.1, this morphism factors as a composition $\Delta ^1 \xrightarrow {v} \operatorname{\mathcal{E}}' \xrightarrow {q} \operatorname{\mathcal{E}}$, where $v$ is anodyne and $q$ is a Kan fibration. Since $\operatorname{\mathcal{E}}$ is a Kan complex and $q$ is a Kan fibration, the simplicial set $\operatorname{\mathcal{E}}'$ is a Kan complex (Remark 3.1.1.11). Because $\Delta ^1$ is weakly contractible and $v$ is a weak homotopy equivalence, the Kan complex $\operatorname{\mathcal{E}}'$ is contractible. We can then write $u = F'(v)$ where $F' = F \circ q$. $\square$

Corollary 4.4.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. The following conditions are equivalent:

$(1)$

The objects $X$ and $Y$ are isomorphic.

$(2)$

There exists a connected Kan complex $\operatorname{\mathcal{E}}$, a pair of vertices $\overline{X}, \overline{Y} \in \operatorname{\mathcal{E}}$, and a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satsifying $f( \overline{X}) = X$ and $f( \overline{Y} ) = Y$.

$(3)$

There exists a connected Kan complex $\operatorname{\mathcal{E}}$, a pair of vertices $\overline{X}, \overline{Y} \in \operatorname{\mathcal{E}}$, and a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satsifying $f( \overline{X}) = X$ and $f( \overline{Y} ) = Y$.

Notation 4.4.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We let $\pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ denote the set of connected components of the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$. Note that $\pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ can be identified with the set of isomorphism classes of objects of $\operatorname{\mathcal{C}}$ (that is, the quotient of the set of objects of $\operatorname{\mathcal{C}}$ by the equivalence relation of isomorphism).

If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$ can be characterized by a universal property:

Proposition 4.4.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a Kan complex. Then composition with the inclusion $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{C}})$.

Proof. Let $F: X \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. To show that $F$ factors through the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$, we must show that for every edge $u: x \rightarrow y$ of the Kan complex $X$, the image $F(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$. This follows from Remark 1.4.1.6, since $u$ is automatically an isomorphism in the $\infty $-category $X$ (Proposition 1.3.6.11). $\square$

Corollary 4.4.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is the largest Kan complex which is contained in $\operatorname{\mathcal{C}}$.

Corollary 4.4.3.14 (Pullbacks of Isofibrations). Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r]^-{F} \ar [d]^{q'} & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \operatorname{\mathcal{D}}' \ar [r]^-{F'} & \operatorname{\mathcal{D}}, } \]

where $q$ is an isofibration of $\infty $-categories and $\operatorname{\mathcal{D}}'$ is an $\infty $-category. Then:

$(1)$

The simplicial set $\operatorname{\mathcal{C}}'$ is an $\infty $-category.

$(2)$

The diagram of Kan complexes

4.14
\begin{equation} \begin{gathered}\label{equation:pullback-of-core} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'^{\simeq } \ar [r] \ar [d]^{q'^{\simeq }} & \operatorname{\mathcal{C}}^{\simeq } \ar [d]^{q^{\simeq }} \\ \operatorname{\mathcal{D}}'^{\simeq } \ar [r] & \operatorname{\mathcal{D}}^{\simeq } } \end{gathered} \end{equation}

is a pullback square and a homotopy pullback square.

$(3)$

A morphism $u: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}'$ is an isomorphism if and only if $F(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ and $q'(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}'$.

$(4)$

The morphism $q'$ is an isofibration of $\infty $-categories.

Proof. Since $q$ is an isofibration, it is an inner fibration. It follows that the morphism $q'$ is also an inner fibration (Remark 4.1.1.5). Since $\operatorname{\mathcal{D}}'$ is an $\infty $-category, the simplicial set $\operatorname{\mathcal{C}}'$ is also an $\infty $-category (Remark 4.1.1.9). This proves $(1)$.

Let $\operatorname{\mathcal{E}}$ denote the fiber product $\operatorname{\mathcal{C}}^{\simeq } \times _{ \operatorname{\mathcal{D}}^{\simeq } } \operatorname{\mathcal{D}}'^{\simeq }$, which we regard as a simplicial subset of $\operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$. It follows from Proposition 4.4.3.6 that $q$ restricts to a Kan fibration $q^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$. The projection map $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}'^{\simeq }$ is a pullback of $q^{\simeq }$, and is therefore also a Kan fibration. Since $\operatorname{\mathcal{D}}'^{\simeq }$ is a Kan complex (Corollary 4.4.3.7), it follows that $\operatorname{\mathcal{E}}$ is a Kan complex (Remark 3.1.1.11). Applying Corollary 4.4.3.13, we deduce that $\operatorname{\mathcal{E}}$ is contained in the core $\operatorname{\mathcal{C}}'^{\simeq } \subseteq \operatorname{\mathcal{C}}'$, which proves that the diagram (4.14) is a pullback square. Since $q^{\simeq }$ is a Kan fibration, it is also a homotopy pullback square (Example 3.4.1.5). This proves assertion $(2)$, and assertion $(3)$ is an immediate consequence.

To complete the proof of $(4)$, it will suffice to show that the morphism $q'$ satisfies condition $(\ast )$ of Definition 4.4.1.4. Let $Y'$ be an object of $\operatorname{\mathcal{C}}'$ and let $\overline{u}': \overline{X}' \rightarrow q'(Y')$ be an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}'$; we wish to show that $\overline{u'}$ can be written as $q'( u')$ for some isomorphism $u': X' \rightarrow Y'$ in the $\infty $-category $\operatorname{\mathcal{C}}'$. By virtue of $(3)$, this is equivalent to showing that $F'(\overline{u}')$ can be written as $q(u)$ for some isomorphism $u: X \rightarrow F(Y)$ in the $\infty $-category $\operatorname{\mathcal{C}}$, which follows from our assumption that $q$ is an isofibration. $\square$

Corollary 4.4.3.15. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories, and let $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ be the fiber of $q$ over an object $D \in \operatorname{\mathcal{D}}$.Then the canonical map $(\operatorname{\mathcal{C}}_{D})^{\simeq } \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}^{\simeq }} \operatorname{\mathcal{C}}^{\simeq }$ is an isomorphism. In other words, the inclusion functor $\operatorname{\mathcal{C}}_{D} \hookrightarrow \operatorname{\mathcal{C}}$ is conservative.

Proof. Apply Corollary 4.4.3.14 in the special case $\operatorname{\mathcal{D}}' = \{ D\} $. $\square$

We close this section by establishing a relative version of Proposition 4.4.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $X$ be an arbitrary simplicial set. Then the simplicial set $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 1.4.3.7), and the simplicial set $\operatorname{Fun}( X, \operatorname{\mathcal{C}}^{\simeq } )$ is a Kan complex (Corollary 3.1.3.4). The inclusion $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces a monomorphism of simplicial sets $\operatorname{Fun}(X, \operatorname{\mathcal{C}}^{\simeq } ) \hookrightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$, which automatically factors through the core $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq }$ (Corollary 4.4.3.13).

Proposition 4.4.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a Kan complex. Then the canonical map

\[ \theta : \operatorname{Fun}(X, \operatorname{\mathcal{C}}^{\simeq } ) \hookrightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}})^{\simeq } \]

is an isomorphism of simplicial sets.

Remark 4.4.3.17. Proposition 4.4.3.12 can be regarded as a special case of Proposition 4.4.3.16: it is equivalent to the assertion that, for every $\infty $-category $\operatorname{\mathcal{C}}$ and every Kan complex $X$, the canonical map $\operatorname{Fun}(X, \operatorname{\mathcal{C}}^{\simeq } ) \hookrightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}})^{\simeq }$ is bijective on vertices.

Proof of Proposition 4.4.3.16. Let $\sigma : Y \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq }$ be a morphism of simplicial sets, which we identify with a diagram $F: X \times Y \rightarrow \operatorname{\mathcal{C}}$. To show that $\sigma $ factors through the monomorphism $\theta $, it will suffice to show that $F$ factors through the core $\operatorname{\mathcal{C}}^{\simeq } \subseteq \operatorname{\mathcal{C}}$. Equivalently, we wish to show that for every edge $(u,v): (x,y) \rightarrow (x',y')$ in the product simplicial set $X \times Y$, the morphism $F(u,v): F(x,y) \rightarrow F(x',y')$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$. Note that $F(u,v)$ can be identified with a composition of morphisms

\[ F(x,y) \xrightarrow { F(u,\operatorname{id}_ y)} F(x', y) \xrightarrow { F( \operatorname{id}_{x'}, v) } F(x',y') \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. Since the collection of isomorphisms in $\operatorname{\mathcal{C}}$ is closed under composition (Remark 1.3.6.3), it will suffice to show that $F(u,\operatorname{id}_{y} )$ and $F( \operatorname{id}_{x'}, v)$ are isomorphsms in $\operatorname{\mathcal{C}}$. In the first case, this follows from Proposition 4.4.3.12 (applied to the morphism $F|_{ X \times \{ y\} }$), since $X$ is a Kan complex. In the second case, it follows from our assumption that $\sigma $ factors through the core $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \subseteq \operatorname{Fun}(X,\operatorname{\mathcal{C}})$ (and therefore carries the edge $v: y \rightarrow y'$ to an isomorphism in the diagram $\infty $-category $\operatorname{Fun}(X,\operatorname{\mathcal{C}})$). $\square$