I began to consider the following chain of ideas: Consider a Category $\text{Cat} \ X$ of objects of a type $X$. One can view the Category as a graph (objects are nodes and morphisms as directed edges) and thus it's natural to ask about topological properties of these graphs.
The simplest example I came up with was to let $X$ be $\mathbb{N}$, and let the arrows map $a \rightarrow b$ in the natural way if $b|a$.
Does $\text{Cat} \ \mathbb{N}$ then have finite genus?
Some Analysis:
There is one true sink "1", the 2nd order sinks are prime numbers, 3rd order sinks are semiprimes, etc...
Note that the set of powers of $2$, after forgetting the direction of the arrows, forms a copy of the complete graph on countably infinitely many vertices, $K_\omega$. This means the genus is in fact infinite.