I am delving a bit into category theory and something has me curious about opposite categories. I have checked several books and I can't seem to find an answer.
Given a category C, the opposite category is just the abstract category with the objects of C and with the arrows of C reversed. However, the opposite category can sometimes be realised (is equivalent to) a category where the objects are sets with additional structure, and the arrows are homomorphisms of these structures.
For instance one of the examples on Wikipedia is that the opposite category of commutative rings is equivalent to the category of affine schemes.
Question. How would one in general, given a category C, find a category where the objects are mathematical structures with underlying sets that satisfy additional axioms, and the morphisms are homomorphisms of those structures, and which is the opposite of the original category C?
For instance, if we take the category where the objects are groups, and the morphisms are group homomorphisms, what is its opposite category in the above sense? Is there some way to find this from the first-order axiomatization of groups?
Best Answer
One way of formalizing the desired categories is given by concrete categories. A category is called concrete (more precise: "concretizable") if it has a faithful functor to the category of sets $Set$. Thus the question is: Is the dual of a concrete category concrete again? The answer is yes: Since a composition of faithful functors is faithful and a dual of a faithful functor is also faithful, it suffices to show that $Set^{op}$ is concrete. But it is not hard to see that the contravariant hom-functor $Hom(-,2)$ (i.e. the power set) yields the desired faithful functor $Set^{op} \to Set$.
However, this solution is somewhat useless. If we apply the proof to $Grp^{op}$, we get sets of the form $P(X)$, where $X$ is a group and morphisms $P(X) \to P(Y)$ should be induced by group morphisms $Y \to X$.
Perhaps we should demand of our concretization that it does not reuses the given category? But this seems to be hard to formalize. Anyway, in the category of groups it would be interesting ...
EDIT: What about the following: If $k$ is a field, then $(k-Vect)^{op}$ is equivalent to the category of pairs $(X,p)$, where $X$ is an affine $k$-scheme and $p$ is a rational point of $X$ such that the corresponding maximal ideal $a$ satisfies $a^2=0$. :-) If $R$ is a ring, then $(R-Mod)^{op}$ is equivalent to the category of pairs $(X,p)$, where $X$ is an affine $R$-scheme, $p$ is a $R$-valued point of $X$ such that the closed image of $X \to Spec(R)$ is $Spec(R)$ and and the closed image of $p : Spec(R) \to X$ is cut out by an ideal $a \subseteq \mathcal{O}_X(X)$ with $a^2=0$. What about dropping the affine-condition, do we get "global modules"? Abstract-Nonsense!