The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.

Why was there the necessity of singling out a particular kind of relations, namely the functional ones? I guess (but I don't have data about this) historically the recognition that "operational" expressions like $x^3$ or $\sum_{i=0}^{\infty} \frac{x^n}{n!}$ could be formalized as functional relations led to devote more attention to functions understood in the modern set theoretical sense (i.e. as a special case of relations). That viewpoint permitted to consider things such as the Dirichlet function $\chi_{\mathbb{Q}}$ (which was previously not even considered to be a true "function"!) as fully legitimate objects, and to not dismiss them as pathological, with great theoretical advantage. The language and notation of functions was preferred even to deal with things that, technically, were relations: think of "multi-valued functions" in complex analysis such as $\sqrt x$ or $\log (x)$.

1) In which instances in modern mathematics are relations used as important generalizations of functions? One example that comes to mind is

correspondencesin the sense of algebraic geometry.

In modern Algebra the concept of homomorphism, a kind of function between algebraic structures, is central; we are used to see expressions like $f(x*y)=f(x)*f(y)$. But it would be equally possible to define a "homomorphic relation" $R$, for example on groups, by the requirement: $(xRz$ & $yRt)$ $\Rightarrow$ $(x*y)R(z*t)$, where $*$ is the group multiplication.

2) Has this kind of "homomorphic relations" been studied (on groups or other algebraic structures)? Why algebra is pervaded with homomorphisms but we never see "homomorphic relations"? Are there something more than just historical reasons?

Let **Set** be the usual category of sets, and **Rel** be the category of sets-with-relations-as-morphisms.

There is the faithful functor **Set** $\to$ **Rel** that simply keeps sets intact and sends a function to its graph. And there is also a faithful functor **Rel** $\to$ **Set** mapping $X\to 2^X$ and $R\subseteq X\times Y$ to $R_*:2^X\to 2^Y, A\mapsto R_*(A)=\{ y\in Y\; |\; \exists x \in A : (x,y)\in R \}$.

Despite the trivial foundational fact that set theoretical functions are *defined* to be a special kind of relations, it seems that in category theory **Set** has priority on **Rel**. For example the Yoneda's lemma is stated for **Set**; and people talk of simplicial *sets*, not simplicial relations; and the category **Rel** is just retrieved as "the Kleisli category of the powerset endofunctor on **Set**" (I just learned this from wikipedia) and it doesn't seem to be so ubiquitous as **Set** (but this impression might just depend on my ignorance in category theory).

3) Are functions really more central/important than relations in category theory? If so, is it just for historical reasons or there are some more "intrinsic" reasons? E.g. is there an analogous of Yoneda's lemma for

Rel?

totalfunctions instead ofpartialfunctions. I think the answer to both questions is the same, however. $\endgroup$sheaf theory. If the "sheaf of local solutions" of a differential operator is taken as a subsheaf of the sheaf of germs of continuous/smooth/analytic functions, it allows you to express that the solutions vary continuously/smoothy/analytically etc. $\endgroup$11more comments