In topology a cover of a space is a set of subspaces whose union is the space. Obviously a subspace is an inclusion.
A space is an object in the category $Top$.
In category theory, according to nlab, a cover of an object $U$ (of some category) is a family of morphisms $f_i:U_i\rightarrow U$. It doesn't look as though they require that the 'union' of the $U_i$ needs to be $U$. Nor that they should be 'inclusions'. But if we do require it, what is the best way to generalise the condition? Is there standard nomenclature for this?
The inclusion of a subspace $U_i$, is probably best generalised by requiring that each $f_i$ is monic.
For the 'union' of the $U_i$, can we simply say that the maps are jointly epic? That is whenever $g\circ f_i=h\circ f_i$ for all $i$, we have $g=h$. But this simply says that $U$ is a coproduct of the family $f_i$.
Or alternatively, in topology, we can take the disjoint union of the cover and this will be surjective and locally injective - that is a local homeomorphism. And this is also called an etale mapping.
If the category has all pushouts and an initial object $0$, we can construct the disjoint pushout $\sqcup g_i$ by the initial morphisms $g_i:0\rightarrow U_i$, and then by the pushout property we have $\sqcup g_i:\sqcup U_i\rightarrow U$ that is epic. But in the presence of initial objects pushouts & coproducts can construct each other. So the alternative is actually no real alternative.
