Item 4 (00ZR) — The Clowder Project

11.4.4 The Core of a Category

Let $\mathcal{C}$ be a category.

Definition 11.4.4.1.1 ▶ The Core of a Category

The core of $\mathcal{C}$ is the pair $(\mathsf{Core}(\mathcal{C}),\iota _{\mathcal{C}})$ consisting of

•
A groupoid $\mathsf{Core}(\mathcal{C})$;
•
A functor $\iota _{\mathcal{C}}\colon \mathsf{Core}(\mathcal{C})\hookrightarrow \mathcal{C}$;

satisfying the following universal property:

•
Given another such pair $(\mathcal{G},i)$, there exists a unique functor $\mathcal{G}\overset {\exists !}{\to }\mathsf{Core}(\mathcal{C})$ making the diagram
commute.

Notation 11.4.4.1.2 ▶ Alternative Notation for the Core of a Category

We also write $\mathcal{C}^{\simeq }$ for $\mathsf{Core}(\mathcal{C})$.

Construction 11.4.4.1.3 ▶ Construction of the Core of a Category

The core of $\mathcal{C}$ is the wide subcategory of $\mathcal{C}$ spanned by the isomorphisms of $\mathcal{C}$, i.e. the category $\mathsf{Core}(\mathcal{C})$ where¹

1.
Objects. We have

\[ \operatorname {\mathrm{Obj}}(\mathsf{Core}(\mathcal{C})) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Obj}}(\mathcal{C}). \]
2.
Morphisms. The morphisms of $\mathsf{Core}(\mathcal{C})$ are the isomorphisms of $\mathcal{C}$.

¹Slogan: The groupoid $\mathsf{Core}(\mathcal{C})$ is the maximal subgroupoid of $\mathcal{C}$.

Proof of Construction 11.4.4.1.3.

This follows from the fact that functors preserve isomorphisms (Item 1 of Proposition 11.5.1.1.6).

Proposition 11.4.4.1.4 ▶ Properties of the Core of a Category

Let $\mathcal{C}$ be a category.

1.
Functoriality. The assignment $\mathcal{C}\mapsto \mathsf{Core}(\mathcal{C})$ defines a functor

\[ \mathsf{Core}\colon \mathsf{Cats}\to \mathsf{Grpd}. \]
2.
2-Functoriality. The assignment $\mathcal{C}\mapsto \mathsf{Core}(\mathcal{C})$ defines a 2-functor

\[ \mathsf{Core}\colon \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Grpd}_{\mathsf{2}}. \]
3.
Adjointness. We have an adjunction
witnessed by a bijection of sets

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{G},\mathcal{D})\cong \operatorname {\mathrm{Hom}}_{\mathsf{Grpd}}(\mathcal{G},\mathsf{Core}(\mathcal{D})), \]

natural in $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$ and $\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, forming, together with the functor $\mathrm{K}_{0}$ of Item 1 of Proposition 11.4.3.1.3, a triple adjunction
witnessed by bijections of sets

\begin{align*} \operatorname {\mathrm{Hom}}_{\mathsf{Grpd}}(\mathrm{K}_{0}(\mathcal{C}),\mathcal{G}) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},\mathcal{G}),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{G},\mathcal{D}) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Grpd}}(\mathcal{G},\mathsf{Core}(\mathcal{D})),\end{align*}

natural in $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$.

4.
2-Adjointness. We have an adjunction
witnessed by an isomorphism of categories

\[ \mathsf{Fun}(\mathcal{G},\mathcal{D})\cong \mathsf{Fun}(\mathcal{G},\mathsf{Core}(\mathcal{D})), \]

natural in $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$ and $\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, forming, together with the 2-functor $\mathrm{K}_{0}$ of Item 2 of Proposition 11.4.3.1.3, a triple 2-adjunction
witnessed by isomorphisms of categories

\begin{align*} \mathsf{Fun}(\mathrm{K}_{0}(\mathcal{C}),\mathcal{G}) & \cong \mathsf{Fun}(\mathcal{C},\mathcal{G}),\\ \mathsf{Fun}(\mathcal{G},\mathcal{D}) & \cong \mathsf{Fun}(\mathcal{G},\mathsf{Core}(\mathcal{D})),\end{align*}

natural in $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$.

Symmetric Strong Monoidality With Respect to Products. The core functor of Item 1 has a symmetric strong monoidal structure

\[ (\mathsf{Core},\mathsf{Core}^{\times },\mathsf{Core}^{\times }_{\mathbb {1}}) \colon (\mathsf{Cats},\times ,\mathsf{pt}) \to (\mathsf{Grpd},\times ,\mathsf{pt}) \]

being equipped with isomorphisms

\[ \begin{gathered} \mathsf{Core}^{\times }_{\mathcal{C},\mathcal{D}} \colon \mathsf{Core}(\mathcal{C})\times \mathsf{Core}(\mathcal{D}) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Core}(\mathcal{C}\times \mathcal{D}),\\ \mathsf{Core}^{\times }_{\mathbb {1}} \colon \mathsf{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Core}(\mathsf{pt}), \end{gathered} \]

natural in $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$.

Symmetric Strong Monoidality With Respect to Coproducts. The core functor of Item 1 has a symmetric strong monoidal structure

\[ (\mathsf{Core},\mathsf{Core}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}},\mathsf{Core}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathbb {1}}) \colon (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}) \to (\mathsf{Grpd},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}) \]

being equipped with isomorphisms

\[ \begin{gathered} \mathsf{Core}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathcal{C},\mathcal{D}} \colon \mathsf{Core}(\mathcal{C})\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathsf{Core}(\mathcal{D}) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Core}(\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}),\\ \mathsf{Core}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathbb {1}} \colon \text{Ø}_{\mathsf{cat}}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Core}(\text{Ø}_{\mathsf{cat}}), \end{gathered} \]

natural in $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$.

Proof of Proposition 11.4.4.1.4.

Item 1: Functoriality

Omitted.

Item 2: 2-Functoriality

Omitted.

Item 3: Adjointness

Omitted.

Item 4: 2-Adjointness

Omitted.

Item 5: Symmetric Strong Monoidality With Respect to Products

Omitted.