Fixed a connected topological space $X$ it's an exercise to show that, if $X$ admits a universal covering $Y \rightarrow X$, then the category $C$ of finite covering spaces of $X$ is small.
I'm interested in the converse proposition, that is: "if $C$ is small then $X$ admits a universal covering?". I strongly suspect this is true, someway I can think about the universal covering as a "limitator" for the category. But I have no idea on how use the smallness of $C$.
I thought about the problem studying Galois Categories, where our $C$ is a classical example. This means that it has a list of good properties:
- Initial and final object
- There exists finite sums, fibre products, finite projective limits, quotients by finite groups of automorphisms.
- Every morphism factorize as the composition of an epimorphism and a monomorphism.