For a path-connected space $X$, a covering space is a fiber bundle with a discrete set. It is known that if $X$ in addition locally path-connected and semilocally simply-connected, then $X$ has a universal cover.
Going over the construction of the universal cover, I see that it is the fibrant replacement of the inclusion of a point $x_0\hookrightarrow X$, i.e. replacing $x_0$ with the mapping path space $P(X,x_0)$ which is contractible and fibrates onto $X$.
I wonder what if we dropped the condition that $X$ is locally path-connected and semilocally simply-connected, then wouldn't the fibration $P(X,x_0)\to X$ still be very interesting? It will not be a universal cover per se, but will be a homotopy version of it in the sense of a fibration from a contractible space. What good properties still can be said for $P(X,x_0)$ and what are lost?
The path space in the path space fibration is not the universal cover, even up to homotopy, unless $X$ is aspherical. Its fiber, rather than being the fundamental group, is the based loop space of $X$. But it is reasonable to think of it as a higher analogue of the universal cover.