The set of connected components of $\mathcal{C}$ is the set $\pi _{0}\webleft (\mathcal{C}\webright )$ whose elements are the connected components of $\mathcal{C}$.
-
1.
Functoriality. The assignment $\mathcal{C}\mapsto \pi _{0}\webleft (\mathcal{C}\webright )$ defines a functor
\[ \pi _{0} \colon \mathsf{Cats}\to \mathsf{Sets}. \] -
2.
Adjointness. We have a quadruple adjunction
-
3.
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
.
-
4.
Preservation of Colimits. The functor $\pi _{0}$ of Item 1 preserves colimits. In particular, we have bijections of sets
\[ \begin{gathered} \begin{aligned} \pi _{0}\webleft (\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}\webright ) & \cong \pi _{0}\webleft (\mathcal{C}\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\pi _{0}\webleft (\mathcal{D}\webright ),\\ \pi _{0}\webleft (\mathcal{C}\mathbin {\textstyle \coprod _{\mathcal{E}}}\mathcal{D}\webright ) & \cong \pi _{0}\webleft (\mathcal{C}\webright )\mathbin {\textstyle \coprod _{\pi _{0}\webleft (\mathcal{E}\webright )}}\pi _{0}\webleft (\mathcal{D}\webright ), \end{aligned} \\ \pi _{0}\webleft (\operatorname {\mathrm{CoEq}}\webleft (\mathcal{C}\underset {G}{\overset {F}{\rightrightarrows }}\mathcal{D}\webright )\webright ) \cong \operatorname {\mathrm{CoEq}}\webleft (\pi _{0}\webleft (\mathcal{C}\webright )\underset {\pi _{0}\webleft (G\webright )}{\overset {\pi _{0}\webleft (F\webright )}{\rightrightarrows }}\pi _{0}\webleft (\mathcal{D}\webright )\webright ), \end{gathered} \]natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$.
-
5.
Symmetric Strong Monoidality With Respect to Coproducts. The connected components functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\pi _{0},\pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0},\pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Cats},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}_{\mathsf{cat}}\webright ) \to \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}\webright ), \]being equipped with isomorphisms
\[ \begin{gathered} \pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathcal{C},\mathcal{D}} \colon \pi _{0}\webleft (\mathcal{C}\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\pi _{0}\webleft (\mathcal{D}\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\pi _{0}\webleft (\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}\webright ),\\ \pi ^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{0|\mathbb {1}} \colon \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\pi _{0}\webleft (\text{Ø}_{\mathsf{cat}}\webright ), \end{gathered} \]natural in $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$.
-
6.
Symmetric Strong Monoidality With Respect to Products. The connected components functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\pi _{0},\pi ^{\times }_{0},\pi ^{\times }_{0|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Cats},\times ,\mathsf{pt}\webright ) \to \webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright ), \]being equipped with isomorphisms
\[ \begin{gathered} \pi ^{\times }_{0|\mathcal{C},\mathcal{D}} \colon \pi _{0}\webleft (\mathcal{C}\webright )\times \pi _{0}\webleft (\mathcal{D}\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\pi _{0}\webleft (\mathcal{C}\times \mathcal{D}\webright ),\\ \pi ^{\times }_{0|\mathbb {1}} \colon \mathrm{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\pi _{0}\webleft (\mathsf{pt}\webright ), \end{gathered} \]natural in $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$.
11.3.2.2 Sets of Connected Components of Categories
Let $\mathcal{C}$ be a category.
Let $\mathcal{C}$ be a category.
Proof of Proposition 11.3.2.2.2.
Item 1: Functoriality
Clear.
Item 2: Adjointness
This is proved in Proposition 11.3.1.1.1.
Item 3: Interaction With Groupoids
Clear.
Item 4: Preservation of Colimits
This follows from Item 2 and of .
Item 5: Symmetric Strong Monoidality With Respect to Coproducts
Clear.
Item 6: Symmetric Strong Monoidality With Respect to Products
Clear.
