Item 2 (00ZB) — 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 $\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\iota _{\mathcal{C}}\webright )$ consisting of

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

satisfying the following universal property:²

•
Given another such pair $\webleft (\mathcal{G},i\webright )$, there exists a unique functor $\mathrm{K}_{0}\webleft (\mathcal{C}\webright )\overset {\exists !}{\to }\mathcal{G}$ 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 $\operatorname {\mathrm{Mor}}\webleft (\mathcal{C}\webright )^{-1}\mathcal{C}$ of $\mathcal{C}$ at the set $\operatorname {\mathrm{Mor}}\webleft (\mathcal{C}\webright )$ 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 $\mathcal{C}\mapsto \mathrm{K}_{0}\webleft (\mathcal{C}\webright )$ defines a functor

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

2.
2-Functoriality. The assignment $\mathcal{C}\mapsto \mathrm{K}_{0}\webleft (\mathcal{C}\webright )$ defines a 2-functor

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

Adjointness. We have an adjunction

witnessed by a bijection of sets

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

natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$ and $\mathcal{G}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Grpd}\webright )$, 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}}\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 a 2-adjunction

witnessed by an isomorphism of categories

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

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

Interaction With Classifying Spaces. We have an isomorphism of groupoids

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

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

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

being equipped with isomorphisms

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

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

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

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

being equipped with isomorphisms

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

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

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 .