Recall that a category C is *small* if the class of its morphisms is a set; otherwise, it is *large*. One of many examples of a large category is **Set**, for Russell's paradox reasons. A category C is *locally small* if the class of morphisms between any two of its objects is a set. Of course, a small category is necessarily locally small. The converse is not true, as **Set** is a counterexample.


Now, I can construct categories that are not locally small. However, what's the most common or most reasonable such category?



What's a reasonable category that is not locally small?
The category of multi-spans spans (thanks to everyone below for correcting my terminology). The objects are sets, and a map from $A$ to $B$ is a set $X$ equipped with a map $X → A × B$. The composition of $X → A × B$ and $Y → B × C$ is $X ×\_B Y → A × C$. 


I am stealing notation from algebraic geometry here: $X ×\_B Y$ is the limit of the diagram $X → B ← Y$.


Admittedly, I've never wanted to allow $X$ to be an arbitrary set. I usually want it to be something like a finite set, a finite simplicial complex or a scheme of finite type. But it is certainly natural to define the category without any restrictions.


