The Clowder Project

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

1. 1.

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

   \[ \pi _{0} \colon \mathsf{Cats}\to \mathsf{Sets}. \]
2. 2.

   Adjointness. We have a quadruple adjunction

   

3. 3.

   Interaction With Groupoids. If $\mathcal{C}$ is a groupoid, then we have an isomorphism of categories

   \[ \pi _{0}(\mathcal{C})\cong \mathrm{K}(\mathcal{C}), \]

   where $\mathrm{K}(\mathcal{C})$ is the set of isomorphism classes of $\mathcal{C}$ of .

4. 4.

   Preservation of Colimits. The functor $\pi _{0}$ of Item 1 preserves colimits. In particular, we have bijections of sets

   \[ \begin{gathered} \begin{aligned} \pi _{0}(\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}) & \cong \pi _{0}(\mathcal{C})\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\pi _{0}(\mathcal{D}),\\ \pi _{0}(\mathcal{C}\mathbin {\textstyle \coprod _{\mathcal{E}}}\mathcal{D}) & \cong \pi _{0}(\mathcal{C})\mathbin {\textstyle \coprod _{\pi _{0}(\mathcal{E})}}\pi _{0}(\mathcal{D}), \end{aligned} \\ \pi _{0}(\operatorname {\mathrm{CoEq}}(\mathcal{C}\underset {G}{\overset {F}{\rightrightarrows }}\mathcal{D})) \cong \operatorname {\mathrm{CoEq}}(\pi _{0}(\mathcal{C})\underset {\pi _{0}(G)}{\overset {\pi _{0}(F)}{\rightrightarrows }}\pi _{0}(\mathcal{D})), \end{gathered} \]

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

5. 5.

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

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

   being equipped with isomorphisms

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

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

6. 6.

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

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

   being equipped with isomorphisms

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

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