Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects.

Among extensive categories, the (free) coproduct cocompletions are characterized as those for which every object admits a presentation as a disjoint union of connected objects. Since presentation by coproduct is unique up to isomorphism in extensive categories, this suggests a notion of connected components of an object, defined up to isomorphism.

Can 'connected components' be made functorial? I am not even sure what its codomain would be; $\pi_0:\mathsf{Fam}(\mathsf A)\longrightarrow \mathsf{Fam}(\mathsf A)$ does not look right to me.

An example I have in mind is affine schemes. Here, disconnectedness is measured by the presence of idempotents, so we can (and do) define a functor from all affine schemes into boolean algebras (of the idempotents of their function rings). However, some affine schemes are totally disconnected, so their connected components can not reconstruct them.

This example shows a case in which connected components are always defined (like the category of topological spaces), but can't always reconstruct the original object. So should a connected components functor really be defined on any extensive category?

Will it be part of an adjunction?

Added. I'm having trouble with understanding the unit. Here's the calculation verifying the adjunction $\pi_0\dashv H$ where $H$ is the copower functor. \begin{aligned}\mathsf{Hom}_{\mathsf{C}}\left(X,HA\right) & \cong\mathsf{Hom}_{\mathsf{C}}\left(\coprod_{i\in\Pi_{0}(X)}C_{i},HA\right)\cong\prod_{i\in\Pi_{0}(X)}\mathsf{Hom}_{\mathsf{C}}\left(C_{i},HA\right)\\ & \cong\prod_{i\in\Pi_{0}(X)}\mathsf{Hom}_{\mathsf{C}}\left(C_{i},\coprod_{A}{\bf 1}\right)\cong\prod_{i\in\Pi_{0}(X)}\left(\coprod_{A}\mathsf{Hom}_{\mathsf{C}}(C_{i},{\bf 1})\right)\\ & \cong\prod_{i\in\Pi_{0}(X)}\mathsf{Hom}_{\mathsf{Set}}({\bf 1},A)\cong\mathsf{Hom}_{\mathsf{Set}}\left(\coprod_{i\in\Pi_{0}(X)}{\bf 1},A\right)\\ & \cong\mathsf{Hom}_{\mathsf{Set}}(\Pi_{0}(X),A) \end{aligned}

The key isomorphism seems to be $$\coprod_A\mathsf{Hom}_\mathsf{C}(C_i,\mathbf{1})\cong\mathsf{Hom}_\mathsf{Set}(\mathbf 1,A)$$ which identifies the images individual connected components along $f:\pi_0X\rightarrow A$ with the constant maps associated to each connected component.

The reverse isomorphism $\Phi:\mathsf{Hom}_{\mathsf{Set}}(\Pi_{0}(X),A)\cong \mathsf{Hom}_{\mathsf{C}}\left(X,HA\right)$ is defined analogously, and the unit is defined by the corresponding $\eta _X=\Phi(1_{\pi_0X})$.

I just don't understand how to think of $\eta$ outside of the spatial case. Is there some mantra for what it does, or is the formalism all there is to it, with intuition behind available in the spatial case alone?

• As you say, not every object is a disjoint union of connected objects. How do you even define $\pi_0$ in that case? – Zhen Lin May 17 '16 at 16:37
• @ZhenLin I don't see a way, but I'm glad to see I'm not missing anything. Is there a sensible definition for coproduct completions? – Arrow May 18 '16 at 7:41
• You could define $\pi_0$ as the left adjoint of the functor $\mathbf{Set} \to \mathcal{A}$ that sends $X$ to $\coprod_{x \in X} 1$, at least when you have $1$. – Zhen Lin May 18 '16 at 12:59
• @ZhenLin I was just writing that out for myself, thanks! Looks like it's pretty crucial to be of the form $\mathsf{Fam}(\mathsf A)$, since then $\pi_0$ is the "family fibration", sending a family of objects to its index set. An I correct in this? – Arrow May 18 '16 at 13:01
• In that situation it is clear what $\pi_0$ should be, yes. Otherwise it is not clear. – Zhen Lin May 18 '16 at 13:02

Let $\mathcal{A}$ be a locally small category with a terminal object. Then $\mathbf{Fam} (\mathcal{A})$ is also locally small with a terminal object $1$, so we can define $\Gamma = \mathrm{Hom}(1, -) : \mathbf{Fam} (\mathcal{A}) \to \mathbf{Set}$. This functor has a left adjoint $\Delta : \mathbf{Set} \to \mathbf{Fam} (\mathcal{A})$, defined on objects by $X \mapsto \coprod_{x \in X} 1$. Furthermore, $\Delta$ itself has a left adjoint $\pi_0 : \mathbf{Fam} (\mathcal{A}) \to \mathbf{Set}$, defined by sending each family of objects in $\mathcal{A}$ to its indexing set.
Of course, not all extensive categories are of the form $\mathbf{Fam} (\mathcal{A})$. In those cases it is less clear how to define $\pi_0$. While one can still define connectedness, what is needed is a definition of totally disconnected object. Once we have that, we could potentially define $\pi_0$ as the reflector into the subcategory of totally disconnected objects (if such a reflector exists).
• I'm having trouble making sense of the unit $\eta:1\Rightarrow H\circ \pi_0$. Spatially it looks like it sends a point of a space to its connected component, but I don't see how to define it categorically. Defining it as $\eta_B=\Phi(1_{\pi_0B})$ also seems tautological. Could you explain what $\eta$ is? – Arrow May 18 '16 at 16:19
• I added my calculation verifying the adjunction along with the definition of $\eta$ to the body of the question. I'm hoping for intuition for $\eta$ in the general case. – Arrow May 19 '16 at 7:38