If we have a topological space $(X,T)$, then we can define the category $(X,P)$, where $P$ are morphisms between objects in $X$ such that the homset $Hom(a,b)$ is the set of all continuous (according to $T$) paths between $a$ and $b$ in $X$.
Can we rederive $T$ from $P$? i.e. does the category of continous paths on $X$ completely determine the topological structure on $X$?
If $X$ is totally disconnected (and there are many of those that are topologically distinct, like $\mathbb{Q}$, the irrationals, the Cantor set etc.) all have the property that $\textrm{Hom}(I,X)$ (where $I$ the closed unit interval) is just a set of constant functions. So the paths into $X$ contain no real topological information on $X$ (just its size: the number of constant paths).
Any "paths" from a connected space is not enough information...