How to define a unit and a counit.

Question

We want to prove that the left adjoint to the discrete category functor $\delta:Set\to Cat$ is the connected component functor $\sigma: Cat\to Set$. I am wanting to know how I would define a counit $\eta: id_{Set}\Rightarrow\sigma\delta$ and a unit $\varepsilon:\delta\sigma\Rightarrow id_{Cat}$.Would $\eta_S:S\to \sigma\delta S$ send an element of $S$ to the equivalence class containing that element for all sets $S$? Also, I have no clue how to define the components of the unit.

Answer 1

Just a terminology thing, you've mixed up the counit and unit. One way to remember this is that the terminology comes from monads (and thence from monoids and so on) where there's a unit morphism $1 \to T$. When $\sigma \dashv \delta$, $\delta \sigma$ forms a monad, so there's a unit morphism $1 \to \delta \sigma$.

Incidentally, this might help you understand what to do. What you have right now requires maps $1 \to \sigma \delta$ and $\delta \sigma \to 1$, which is the reverse of what you want. So you'll need a unit $\mathcal{C} \to \delta(\sigma(\mathcal{C}))$ and a counit $\sigma(\delta(S)) \to S$ for categories $\mathcal{C}$ and sets $S$.

Can you make a (natural) map from a category to the discrete category on its connected components? Can you define a map from the connected components of the discrete category on a set to that set?