Item 3 (00ZC) — The Clowder Project

11.4.3 The Groupoid Completion of a Category

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

Definition 11.4.3.1.1 ▶ The Groupoid Completion of a Category

The groupoid completion of $\mathcal{C}$¹ is the pair $(\mathrm{K}_{0}(\mathcal{C}),\iota _{\mathcal{C}})$ consisting of

•
A groupoid $\mathrm{K}_{0}(\mathcal{C})$;
•
A functor $\iota _{\mathcal{C}}\colon \mathcal{C}\to \mathrm{K}_{0}(\mathcal{C})$;

satisfying the following universal property:²

•
Given another such pair $(\mathcal{G},i)$, there exists a unique functor $\mathrm{K}_{0}(\mathcal{C})\overset {\exists !}{\to }\mathcal{G}$ making the diagram
commute.

¹Further Terminology: Also called the Grothendieck groupoid of $\mathcal{C}$ or the Grothendieck groupoid completion of $\mathcal{C}$.
²See Item 5 of Proposition 11.4.3.1.3 for an explicit construction.

Construction 11.4.3.1.2 ▶ Construction of the Groupoid Completion of a Category

Concretely, the groupoid completion of $\mathcal{C}$ is the Gabriel–Zisman localisation $\operatorname {\mathrm{Mor}}(\mathcal{C})^{-1}\mathcal{C}$ of $\mathcal{C}$ at the set $\operatorname {\mathrm{Mor}}(\mathcal{C})$ of all morphisms of $\mathcal{C}$; see , .

(To be expanded upon later on.)

Proof of Construction 11.4.3.1.2.

Omitted.

Proposition 11.4.3.1.3 ▶ Properties of Groupoid Completion

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

1.
Functoriality. The assignment $\mathcal{C}\mapsto \mathrm{K}_{0}(\mathcal{C})$ defines a functor

\[ \mathrm{K}_{0} \colon \mathsf{Cats}\to \mathsf{Grpd}. \]
2.
2-Functoriality. The assignment $\mathcal{C}\mapsto \mathrm{K}_{0}(\mathcal{C})$ defines a 2-functor

\[ \mathrm{K}_{0} \colon \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Grpd}_{\mathsf{2}}. \]

3.
Adjointness. We have an adjunction
witnessed by a bijection of sets

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Grpd}}(\mathrm{K}_{0}(\mathcal{C}),\mathcal{G})\cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},\mathcal{G}), \]

natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$, forming, together with the functor $\mathsf{Core}$ of Item 1 of Proposition 11.4.4.1.4, 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 $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$.

2-Adjointness. We have a 2-adjunction

witnessed by an isomorphism of categories

\[ \mathsf{Fun}(\mathrm{K}_{0}(\mathcal{C}),\mathcal{G})\cong \mathsf{Fun}(\mathcal{C},\mathcal{G}), \]

natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$, forming, together with the 2-functor $\mathsf{Core}$ of Item 2 of Proposition 11.4.4.1.4, 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 $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$.

Interaction With Classifying Spaces. We have an isomorphism of groupoids

\[ \mathrm{K}_{0}(\mathcal{C}) \cong \Pi _{\leq 1}(\left\lvert \mathrm{N}_{\bullet }(\mathcal{C})\right\rvert ), \]

natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$; i.e. the diagram

commutes up to natural isomorphism.

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

\[ (\mathrm{K}_{0},\mathrm{K}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0},\mathrm{K}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathbb {1}}) \colon (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}) \to (\mathsf{Grpd},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}) \]

being equipped with isomorphisms

\[ \begin{gathered} \mathrm{K}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathcal{C},\mathcal{D}} \colon \mathrm{K}_{0}(\mathcal{C})\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathrm{K}_{0}(\mathcal{D}) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{K}_{0}(\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}),\\ \mathrm{K}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathbb {1}} \colon \text{Ø}_{\mathsf{cat}}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{K}_{0}(\text{Ø}_{\mathsf{cat}}), \end{gathered} \]

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

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

\[ (\mathrm{K}_{0},\mathrm{K}^{\times }_{0},\mathrm{K}^{\times }_{0|\mathbb {1}}) \colon (\mathsf{Cats},\times ,\mathsf{pt}) \to (\mathsf{Grpd},\times ,\mathsf{pt}) \]

being equipped with isomorphisms

\[ \begin{gathered} \mathrm{K}^{\times }_{0|\mathcal{C},\mathcal{D}} \colon \mathrm{K}_{0}(\mathcal{C})\times \mathrm{K}_{0}(\mathcal{D}) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{K}_{0}(\mathcal{C}\times \mathcal{D}),\\ \mathrm{K}^{\times }_{0|\mathbb {1}} \colon \mathsf{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{K}_{0}(\mathsf{pt}), \end{gathered} \]

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

Proof of Proposition 11.4.3.1.3.

Item 1: Functoriality

Omitted.

Item 2: 2-Functoriality

Omitted.

Item 3: Adjointness

Omitted.

Item 4: 2-Adjointness

Omitted.

Item 5: Interaction With Classifying Spaces

See Corollary 18.33 of .