I found myself a bit puzzled over the distinction between adjoint functors and inverse functors. They both seem to involve relationships between functors and categories, but I'm not clear on how they fundamentally differ. Could someone help clarify this?
Here's my current understanding:
Inverse Functors: These are pairs of functors that essentially 'undo' each other. For two categories $\mathcal{C}$ and $\mathcal{D}$, if we have functors $$F: \mathcal{C} \rightarrow \mathcal{D} \quad \text{and} \quad G: \mathcal{D} \rightarrow \mathcal{C}, $$ they are inverses if $F \circ G$ and $G \circ F$ are naturally isomorphic to the identity functors on $\mathcal{C}$ and $\mathcal{D}$, respectively. This implies a strong relationship, suggesting that $\mathcal{C}$ and $\mathcal{D}$are essentially the same from a categorical standpoint (categorical equivalence).
Adjoint Functors: Here, the relationship is less about equivalence and more about a correspondence between morphisms in different categories. For functors $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{D} \rightarrow \mathcal{C}$, they are adjoint if there's a natural bijection between morphisms $$\text{Hom}_{\mathcal{D}}(F(C), D) \cong\text{Hom}_{\mathcal{C}}(C, G(D)),\qquad \forall C \in \mathcal{C}, \, \forall D \in \mathcal{D}. $$ This relationship is fundamental in various areas of mathematics and provides a way of translating structures between categories.
My questions are:
- Could you provide some intuitive examples that clearly differentiate between the two?
- In what mathematical contexts are each of these concepts most commonly applied? Any insights or corrections to my understanding would be greatly appreciated!
Categorical equivalence means that the involved categories encode the same informations, even if the objects or the arrows do not correspond each other bijectively.
For example consider two categories: let $\mathcal C$ be the category such that $\operatorname {Ob}\mathcal C = \{C\}$ and $\operatorname{Mor}\mathcal C = \{id_C\}$ and let $\mathcal D$ the category with two objects $\operatorname{Ob}\mathcal D = \{A,B\}$ and four arrows: the identities $id_A$ and $id_B$ plus two arrows $f : A \to B$ and $g :B \to A$ such that their composition is the identity of the right object. So $\mathcal D$ is a category with two distinct objects which are isomorphic. Clearly you can't find two functors $F : \mathcal C \to \mathcal D$, $G : \mathcal D \to \mathcal C$ such that their compositions are the identies of these categories as this would imply a bijection between their objects, so these categories are not isomorphic. But it is clear that these categories encode the same categorical information, as the objects of a category are often described by universal properties, which, by definition, do not distinguish isomorphic objects. That's why you want to weaken the notion of isomorphism to be able to say that categories like mine are "basically, basically the same". Indeed it's not difficult to prove that $\mathcal C$ and $\mathcal D$ are equivalent, just check that the constant functor $F : \mathcal D \to \mathcal C$ is fully-faithful and essentially surjective on objects. Mathematics is plenty of significant examples of equivalent categories, just to cite some you have the equivalence of reductive $k$-algebras and affine varieties over $k$.
The notion of adjointness is much more evanescent. I'm not a category theorist so I don't know if there's a nicer way to explain it, but the existence of a pair of adjoint functors between two categories means that they embed into each other in somehow nice way. The first application one is usually taught is that right-adjoints preserve limits while left-adjoints colimits, so in presence of an adjunction you obtain a strict relations between these objects. For example if $\mathcal C$ is a category admitting a forgetful functor on $\mathsf{Set}$ which is right-adjoint, then you know that products in $\mathcal C$ have the set-structure of the usual cartesian products of sets. Many categories of algebraic structure admit such a functor, like $\mathsf{Grp}$, $R$-$\mathsf{Mod}$. That's why in these categories the product of their object is the cartesian product with some structure. On the contrary, the forgetful functor is rarely left-adjoint, that's why in general colimits of algebraic categories have a wilder behavior.
If you want an explicit example of non-equivalent categories which are related by an adjuction you may think to $\mathsf {Ab}$ and $\mathsf{Grp}$ with the functors: $J : \mathsf{Ab} \to \mathsf{Grp}$ which makes an abelian group forget to be abelian, and $ab : \mathsf{Grp} \to \mathsf{Ab}$ which sends a group in its abelianization. You can check that $ab \dashv J$. However, these functors are not quasi-inverses as $J$ is fully faithful, but not essentially surjective on objects. More is true: there is no hope to find an equivalence between these categories: if $F : \mathsf{Grp} \to \mathsf{Ab}$ were a quasi-inverse between these categories, then $$ F(A \times B) \simeq F(A) \oplus F(B)\\ F(A \ast B) \simeq F(A) \oplus F(B) $$ as it preserves both products and coproducts. This would imply that $A \ast B \simeq A \times B$ for any pair of groups, but this is false in general.