The Clowder Project

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

1. 1. Functoriality. The assignment $\mathcal{C}\mapsto {\pi }_0\left(\mathcal{C}\right)$ defines a functor
   $${\pi }_0:\mathsf{Cats}\to \mathsf{Sets}.$$
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
   $${\pi }_0\left(\mathcal{C}\right)\cong \mathrm{K}\left(\mathcal{C}\right),$$

   where $\mathrm{K}\left(\mathcal{C}\right)$ 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
   $$\begin{array}{c}\begin{array}{rl}{\pi }_0\left(\mathcal{C}\coprod \mathcal{D}\right)& \cong {\pi }_0\left(\mathcal{C}\right)\coprod {\pi }_0\left(\mathcal{D}\right),\\ {\pi }_0\left(\mathcal{C}\coprod _{\mathcal{E}}\mathcal{D}\right)& \cong {\pi }_0\left(\mathcal{C}\right)\coprod _{{\pi }_0\left(\mathcal{E}\right)}{\pi }_0\left(\mathcal{D}\right),\end{array}\\ {\pi }_0\left(\text{CoEq}\left(\mathcal{C}\underset{G}{\stackrel{F}{\rightrightarrows }}\mathcal{D}\right)\right)\cong \text{CoEq}\left({\pi }_0\left(\mathcal{C}\right)\underset{{\pi }_0\left(G\right)}{\stackrel{{\pi }_0\left(F\right)}{\rightrightarrows }}{\pi }_0\left(\mathcal{D}\right)\right),\end{array}$$

   natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\left(\mathsf{Cats}\right)$.

5. 5. Symmetric Strong Monoidality With Respect to Coproducts. The connected components functor of Item 1 has a symmetric strong monoidal structure
   $$({\pi }_0,{\pi }_0^{\coprod },{\pi }_{0|\mathbb{1}}^{\coprod }):(\mathsf{Cats},\coprod ,{\text{Ø}}_{\mathsf{cat}})\to (\mathsf{Sets},\coprod ,\text{Ø}),$$

   being equipped with isomorphisms

   $$\begin{array}{c}{\pi }_{0|\mathcal{C},\mathcal{D}}^{\coprod }:{\pi }_0\left(\mathcal{C}\right)\coprod {\pi }_0\left(\mathcal{D}\right)\stackrel{\sim }{⇢}{\pi }_0\left(\mathcal{C}\coprod \mathcal{D}\right),\\ {\pi }_{0|\mathbb{1}}^{\coprod }:\text{Ø}\stackrel{\sim }{⇢}{\pi }_0\left({\text{Ø}}_{\mathsf{cat}}\right),\end{array}$$

   natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\left(\mathsf{Cats}\right)$.

6. 6. Symmetric Strong Monoidality With Respect to Products. The connected components functor of Item 1 has a symmetric strong monoidal structure
   $$({\pi }_0,{\pi }_0^{\times },{\pi }_{0|\mathbb{1}}^{\times }):(\mathsf{Cats},\times ,\mathsf{pt})\to (\mathsf{Sets},\times ,\text{pt}),$$

   being equipped with isomorphisms

   $$\begin{array}{c}{\pi }_{0|\mathcal{C},\mathcal{D}}^{\times }:{\pi }_0\left(\mathcal{C}\right)\times {\pi }_0\left(\mathcal{D}\right)\stackrel{\sim }{⇢}{\pi }_0(\mathcal{C}\times \mathcal{D}),\\ {\pi }_{0|\mathbb{1}}^{\times }:\text{pt}\stackrel{\sim }{⇢}{\pi }_0\left(\mathsf{pt}\right),\end{array}$$

   natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\left(\mathsf{Cats}\right)$.