The theory of interpolation spaces is one such example. The classical interpolation theorems of M. Riesz, Thorin and Marcinkiewicz and their generalizations by A. Zygmund, E. M. Stein, G. Weiss, and C. Fefferman are basic tools in harmonic analysis. The Riesz-Thorin theorem yields a quick proof of the Hausdorff-Young inequality on the $L^p \to L^{p'}$ boundedness of the Fourier transform, as well as the classic proof of the $L^p$-boundedness of the Hilbert transform by M. Riesz. The Marcinkiewicz interpolation theorem is effectively a generalization of the proof of the Hardy-Littlewood maximal inequality and is a crucial tool in proving the $L^p$-boundedness of a wide range of singular integrals---say, those of convolution type or the Calder'{o}n-Zygmund operators.
The interpolation theorem of Riesz-Thorin and Marcinkiewicz have been generalized substantially by A. Calder'{o}n and J. Peetre, respectively, and modern treatments of the subject---Bergh-L"{o}fstr"{o}m, Bennett-Sharpley, etc.---often utilize category theory to make the notion of interpolation precise. To this end, we need the notion of a Banach couple, which is a pair $(A_0,A_1)$ of Banach spaces such that there exists a Hausdorff topological vector space $X$ such that we have continuous linear embeddings $A_0 \hookrightarrow X$ and $A_1 \hookrightarrow X$. The collection of Banach couples forms a category, whose typical morphism $T:(A_0,A_1) \to (B_0,B_1)$ is a bounded linear operator $T:A_0 + A_1 \to B_0 + B_1$ such that the restrictions $T|_{A_0}$ and $T|_{A_1}$ are bounded linear operators mapping into $B_0$ and $B_1$, respectively. An interpolation space for a Banach couple $(A_0,A_1)$ is a Banach space $A$ such that
- $A_0 \cap A_1 \hookrightarrow A \hookrightarrow A_0 + A_1$;
- if $T:A_0 + A_1 \to A_0 + A_1$ is a bounded linear operator such that $T|_{A_0}:A_0 \to A_1$ and $T|_{A_1}: A_1 \to A_1$ are bounded linear operators, then the restriction of $T$ on $A$ is a bounded linear operator into $A$;
and an interpolation functor a functor from the category of Banach couples to the category of Banach spaces that sends Banach couples to corresponding interpolation spaces and transforms morphisms between Banach couples into the corresponding bounded linear operators between sums of Banach spaces. With this terminology, we can view an interpolation theorem as a construction of such a functor.
Since this definition is quite general, it is conceivable to have interpolation theorems that are not quite as nice as the classical ones. To characterize the nicer ones, we define exact interpolation spaces $A$ and $B$ to be interpolation spaces of Banach couples $(A_0,A_1)$ and $(B_0,B_1)$ such that
$$ \|T\|_{A \to B} \leq \max \left( \|T\|_{A_0 \to B_0} , \|T\|_{A_1 \to B_1} \right)$$
for all morphisms $T: (A_0,A_1) \to (B_0,B_1)$ and an exact interpolation functor an interpolation functor that produces exact interpolation spaces. Now, the theorem of Aronszajin and Gagliardo states that every interpolation space admits an exact interpolation functor that sends the corresponding Banach couple to the interpolation space. With the aid of category-theoretic methods, we thus have a guarantee that our efforts in finding a particular interpolation theorem will not go wasted.