Item 1 (00ZN) — 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 $\webleft (\mathsf{Core}\webleft (\mathcal{C}\webright ),\iota _{\mathcal{C}}\webright )$ consisting of

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

satisfying the following universal property:

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

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 $\mathsf{Core}\webleft (\mathcal{C}\webright )$ where¹

1.
Objects. We have

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

¹Slogan: The groupoid $\mathsf{Core}\webleft (\mathcal{C}\webright )$ 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 $\mathcal{C}\mapsto \mathsf{Core}\webleft (\mathcal{C}\webright )$ defines a functor

\[ \mathsf{Core}\colon \mathsf{Cats}\to \mathsf{Grpd}. \]

2-Functoriality. The assignment $\mathcal{C}\mapsto \mathsf{Core}\webleft (\mathcal{C}\webright )$ defines a 2-functor

\[ \mathsf{Core}\colon \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Grpd}_{\mathsf{2}}. \]

Adjointness. We have an adjunction

witnessed by a bijection of sets

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

natural in $\mathcal{G}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Grpd}\webright )$ and $\mathcal{D}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, 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}}\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\mathcal{G}\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{G}\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{G},\mathcal{D}\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Grpd}}\webleft (\mathcal{G},\mathsf{Core}\webleft (\mathcal{D}\webright )\webright ),\end{align*}

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

2-Adjointness. We have an adjunction

witnessed by an isomorphism of categories

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

natural in $\mathcal{G}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Grpd}\webright )$ and $\mathcal{D}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, 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}\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\mathcal{G}\webright ) & \cong \mathsf{Fun}\webleft (\mathcal{C},\mathcal{G}\webright ),\\ \mathsf{Fun}\webleft (\mathcal{G},\mathcal{D}\webright ) & \cong \mathsf{Fun}\webleft (\mathcal{G},\mathsf{Core}\webleft (\mathcal{D}\webright )\webright ),\end{align*}

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

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

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

being equipped with isomorphisms

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

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

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

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

being equipped with isomorphisms

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

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

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.