Generalized covering space theories

In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are defined purely in terms of unique lifting properties. For each category $\mathcal{C}$ of path-connected spaces having the unit disk as an object, we construct a category of $\mathcal{C}$-coverings over a given space $X$ that embeds in the category of $π_1(X,x_0)$-sets via the usual monodromy action on fibers. When $\mathcal{C}$ is extended to its coreflective hull $\mathscr{H}(\mathcal{C})$, the resulting category of based $\mathscr{H}(\mathcal{C})$-coverings is complete, has an initial object, and often characterizes more of the subgroup lattice of $π_1(X,x_0)$ than traditional covering spaces. We apply our results to three special coreflective subcategories: (1) The category of $Δ$-coverings employs the convenient category of $Δ$-generated spaces and is universal in the sense that it contains every other generalized covering category as a subcategory. (2) In the locally path-connected category, we preserve notion of generalized covering due to Fischer and Zastrow and characterize the topology of such coverings using the standard whisker topology. (3) By employing the coreflective hull $\mathbf{Fan}$ of the category of all contractible spaces, we characterize the notion of continuous lifting of paths and identify the topology of $\mathbf{Fan}$-coverings as the natural quotient topology inherited from the path space.