The Clowder Project

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

1. 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.

   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. 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 $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$.

4. 4.

   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 $\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{Grpd})$.

5. 5.

   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.

6. 6.

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

7. 7.

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