Let $\mathcal{C}$ be a category.
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Functoriality. The assignment $\mathcal{C}\mapsto \pi _{0}\webleft (\mathcal{C}\webright )$ defines a functor
\[ \pi _{0} \colon \mathsf{Cats}\to \mathsf{Sets}. \]
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Adjointness. We have a quadruple adjunction
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Interaction With Groupoids. If $\mathcal{C}$ is a groupoid, then we have an isomorphism of categories
\[ \pi _{0}\webleft (\mathcal{C}\webright )\cong \mathrm{K}\webleft (\mathcal{C}\webright ), \]
where $\mathrm{K}\webleft (\mathcal{C}\webright )$ is the set of isomorphism classes of $\mathcal{C}$ of
.
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Preservation of Colimits. The functor $\pi _{0}$ of Item 1 preserves colimits. In particular, we have bijections of sets
\[ \begin{gathered} \begin{aligned} \pi _{0}\webleft (\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}\webright ) & \cong \pi _{0}\webleft (\mathcal{C}\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\pi _{0}\webleft (\mathcal{D}\webright ),\\ \pi _{0}\webleft (\mathcal{C}\mathbin {\textstyle \coprod _{\mathcal{E}}}\mathcal{D}\webright ) & \cong \pi _{0}\webleft (\mathcal{C}\webright )\mathbin {\textstyle \coprod _{\pi _{0}\webleft (\mathcal{E}\webright )}}\pi _{0}\webleft (\mathcal{D}\webright ), \end{aligned} \\ \pi _{0}\webleft (\text{CoEq}\webleft (\mathcal{C}\underset {G}{\overset {F}{\rightrightarrows }}\mathcal{D}\webright )\webright ) \cong \text{CoEq}\webleft (\pi _{0}\webleft (\mathcal{C}\webright )\underset {\pi _{0}\webleft (G\webright )}{\overset {\pi _{0}\webleft (F\webright )}{\rightrightarrows }}\pi _{0}\webleft (\mathcal{D}\webright )\webright ), \end{gathered} \]
natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.
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Symmetric Strong Monoidality With Respect to Coproducts. The connected components functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\pi _{0},\pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0},\pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset _{\mathsf{cat}}\webright ) \to \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset \webright ), \]
being equipped with isomorphisms
\[ \begin{gathered} \pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathcal{C},\mathcal{D}} \colon \pi _{0}\webleft (\mathcal{C}\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\pi _{0}\webleft (\mathcal{D}\webright ) \xrightarrow {\cong }\pi _{0}\webleft (\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}\webright ),\\ \pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathbb {1}} \colon \emptyset \xrightarrow {\cong }\pi _{0}\webleft (\emptyset _{\mathsf{cat}}\webright ), \end{gathered} \]natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.
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Symmetric Strong Monoidality With Respect to Products. The connected components functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\pi _{0},\pi ^{\times }_{0},\pi ^{\times }_{0|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ) \to \webleft (\mathsf{Sets},\times ,\text{pt}\webright ), \]
being equipped with isomorphisms
\[ \begin{gathered} \pi ^{\times }_{0|\mathcal{C},\mathcal{D}} \colon \pi _{0}\webleft (\mathcal{C}\webright )\times \pi _{0}\webleft (\mathcal{D}\webright ) \xrightarrow {\cong }\pi _{0}\webleft (\mathcal{C}\times \mathcal{D}\webright ),\\ \pi ^{\times }_{0|\mathbb {1}} \colon \text{pt}\xrightarrow {\cong }\pi _{0}\webleft (\mathsf{pt}\webright ), \end{gathered} \]natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.
