How to look at adjunctions correctly?

Question

I am learning some category theory to help me with my area of research. I am trying to get familiar with the notion of adjunction. In some books I see the authors proving that two functors form an adjunction with just a comment, and this is something I find a bit difficult to follow. I will give an example. Suppose that $\mathcal{E}$ is a topos with small colimits and consider the functors $$\Gamma:\mathcal{E}\to \mathbf{Set},\\ \Gamma E=\text{Hom}_{\mathcal{E}}(1,E)$$ and $$\Delta:\mathbf{Set}\to \mathcal{E},\\ \Delta S=\coprod_{s\in S} 1$$ The author says "morphisms $\Delta S\to E$ in $\mathcal{E}$ clearly correspond to functions $S\to \Gamma E$ of sets, so that this functor $\Delta$ is left adjoint to $\Gamma$". I know that this correspondence is what needs to be proved to have an adjunction, but how is it so obvious that the correspondence holds? I have found many situations like this before...It is probably something to do with my mathematical maturity in this area, but any help on how to look correctly at this would be much appreciated.

Answer 1

Such a claim usually means that verifying the details is routine and straightforward (i.e., the steps to follow are clear, and there is no need for anything clever in order to establish the result). With time, you'll learn to agree with such statements and in some case quickly do the proof mentally. For now, if you really wish to understand what is going on, actually carry out the proof.

Answer 2

For this to be obvious, you must know that left adjoints preserve colimits. Given that, simply observe that every set $S$ can be expressed as a coproduct, viz $\coprod_{s \in S} 1$. Coproducts are colimits, so if $\Delta : \mathbf{Set} \to \mathcal{E}$ is a left adjoint, then we must have $\Delta S \cong \coprod_{s \in S} \Delta 1$; and if $\Delta$ is a left adjoint to $\Gamma$, then we must have $$\mathbf{Set} (1, \Gamma E) \cong \mathcal{E} (\Delta 1, E)$$ but $\mathbf{Set} (1, -) \cong \mathrm{id}$, so $$\mathcal{E} (1, E) \cong \mathcal{E} (\Delta 1, E)$$ and therefore $\Delta 1 \cong 1$.

My only advice is this: do not try to learn topos theory without first being comfortable with general category theory.

Answer 3

This isn't really special to topoi. You just have to remember the definition of a coproduct. It implies $$\hom_\mathcal{E}(\Delta S,E)=\prod_{s \in S} \hom(1,E).$$ And by the very definition of a product of sets, this identifies with $$\hom_{\mathsf{Set}}(S,\hom(1,E)).$$ Done. More generally, if $\mathcal{E}$ is any category with coproducts and $X \in \mathcal{E}$ is any object, then $\mathsf{Set} \to \mathcal{E}, ~S \mapsto \coprod_{s \in S} X$ (the copower) is left adjoint to $\hom_\mathcal{E}(X,-) : \mathcal{E} \to \mathsf{Set}$.