Btukfyl

I'm studying category theory by myself and I just came across this sentence from Wikipedia:

An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and D, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

Can someone provide an example of an "upgrade" of an adjunction to an equivalence? I'm interested in understanding why I could intuitively think of an adjunction as a weak form of an equivalence.

Let $mathcal C$ and $mathcal D$ be two categories, and let $Fcolonmathcal Clongrightarrow mathcal D$ and $Gcolonmathcal Dlongrightarrowmathcal C$ be two functors, with $F$ left adjoint to $G$. Then there are natural transformations $Fcirc Glongrightarrow operatorname{Id}_{mathcal D}$ and $operatorname{Id}_{mathcal C}longrightarrow Gcirc F$ (as mentioned in a comment above).

Let $mathcal Asubsetmathcal C$ be the full subcategory consisting of all objects $Ainmathcal C$ for which the natural morphism $Alongrightarrow GF(A)$ is an isomorphism. Similarly, let $mathcal Bsubsetmathcal D$ be the full subcategory consisting of all objects $Binmathcal D$ for which the natural morphism $FG(B)longrightarrow B$ is an isomorphism.

Then one can check that $F(mathcal A)subsetmathcal B$ and $G(mathcal B)subsetmathcal A$. The restrictions of the adjoint functors $F$ and $G$ to the full subcategories $mathcal Asubsetmathcal C$ and $mathcal Bsubsetmathcal D$ are again adjoint functors: the functor $F|_{mathcal A}colon mathcal Alongrightarrowmathcal B$ is left adjoint to the functor $G|_{mathcal B}colon mathcal Blongrightarrowmathcal A$. The adjunction between the functors $F|_{mathcal A}$ and $G|_{mathcal B}$ is an equivalence between the categories $mathcal A$ and $mathcal B$,
$$
F|_{mathcal A}colonmathcal A,simeq,mathcal B:!G|_{mathcal B}.
$$
This result can be found in the paper A. Frankild, P. Jorgensen, "Foxby equivalence, complete modules, and torsion modules", J. Pure Appl. Algebra 174 #2, p.135-147, 2002, https://doi.org/10.1016/S0022-4049(02)00043-9 , Theorem 1.1.

I am not sure whether this should be properly called "upgrading an adjuction to an equivalence", though. The passage from the adjoint pair $(F,G)$ to the equivalence $(F|_{mathcal A},,G|_{mathcal B})$ entails losing rather than gaining information. Perhaps it would be better to call it "restricting an adjunction to an equivalence".

Then again, I do not know what the author of the passage in the Wikipedia article might have had in mind.

Another and probably more natural interpretation of the sentence in the Wikipedia article may be called "localizing an adjunction to an equivalence".

Let $mathcal C$ and $mathcal D$ be two categories, and let $Fcolonmathcal Clongrightarrow mathcal D$ and $Gcolonmathcal Dlongrightarrowmathcal C$ be two functors, with $F$ left adjoint to $G$. Then there are natural transformations $Fcirc Glongrightarrow operatorname{Id}_{mathcal D}$ and $operatorname{Id}_{mathcal C}longrightarrow Gcirc F$, as above.

Denote by $mathcal S$ the multiplicative class of morphisms in $mathcal C$ generated by all the morphisms $Clongrightarrow GF(C)$, where $C$ ranges over the objects of $mathcal C$. Similarly, denote by $mathcal T$ the multiplicative class of morphisms in $mathcal D$ generated by all the morphisms $FG(D)longrightarrow D$, where $D$ ranges over the objects of $mathcal D$.

Then one can check that the composition $mathcal Clongrightarrow mathcal Dlongrightarrow mathcal D[mathcal T^{-1}]$ of the functor $F$ with the localization functor $mathcal Dlongrightarrow mathcal D[mathcal T^{-1}]$ takes all the morphisms from $mathcal S$ to isomorphisms in $mathcal D[mathcal T^{-1}]$. So the functor $Fcolonmathcal Clongrightarrowmathcal D$ descends to a functor $overline Fcolonmathcal C[mathcal S^{-1}]longrightarrowmathcal D[mathcal T^{-1}]$; and similarly the functor $Gcolonmathcal Dlongrightarrowmathcal C$ descends to a functor $overline Gcolonmathcal D[mathcal T^{-1}]longrightarrowmathcal C[mathcal S^{-1}]$.

The functors $overline F$ and $overline G$ are still adjoint to each other, and this adjunction is an equivalence between the two localized categories:
$$
overline Fcolonmathcal C[mathcal S^{-1}],simeq,mathcal D[mathcal T^{-1}]:!overline G.
$$

Yet another and perhaps even more natural interpretation of what may be meant by the sentence in Wikipedia also involves passing to localizations $mathcal C[mathcal S^{-1}]$ and $mathcal D[mathcal T^{-1}]$ of the given two categories $mathcal C$ and $mathcal D$ with respect to some natural multiplicative classes of morphisms (often called the classes of weak equivalences) $mathcal Ssubsetmathcal C$ and $mathcal Tsubsetmathcal D$.

But, rather than hoping that the functors $F$ and $G$ would simply descend to functors between the localized categories, one derives them in some way, producing a left derived functor
$$
mathbb LFcolonmathcal C[mathcal S^{-1}]longrightarrowmathcal D[mathcal T^{-1}]
$$
and a right derived functor
$$
mathbb RGcolonmathcal D[mathcal T^{-1}]longrightarrowmathcal C[mathcal T^{-1}].
$$
Then the functor $mathbb LF$ is usually left adjoint to the functor $mathbb RG$, and under certain assumptions they are even adjoint equivalences.

I would not go into further details on derived functors etc. in this answer, but rather suggest some keywords or a key sentence which you could look up: a Quillen equivalence between two model categories induces an equivalence between their homotopy categories.