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"Fool's covering spaces" are very close to overlays of R. H. Fox (see this paper in the first place and also this one), which I think are still better: they retain all nice properties of "fool's covering spaces" and have additional ones. An equivalent (see "Steenrod homotopy", Lemma 7.3 or Mardesic-Matijevic) definition of an overlay is that it is

"Fool's covering spaces" are very close to overlays of R. H. Fox (see this paper in the first place and also this one), which I think are still better: they retain all nice properties of "fool's covering spaces" and have additional ones. An equivalent (see "Steenrod homotopy", Lemma 7.3 or Mardesic-Matijevic) definition of an overlay is that it is

a map $p:Y\to X$ such that there exists a cover $\{U_\alpha\}$ of $X$ satisfying

 

(i) each $p^{-1}(U_\alpha)=\bigsqcup_\lambda U_\alpha^\lambda$, where each $p$ restricted over $U_\alpha^\lambda$ is a homeomorphism onto $U_\alpha$; and

 

(ii) if $U_\alpha^\lambda\cap U_\beta^\mu$ and $U_\alpha^\lambda\cap U_\beta^\nu$ are both nonempty, then $\mu=\nu$.

a map $p:Y\to X$ such that there exists a cover $\{U_\alpha\}$ of $X$ satisfying

(i) each $p^{-1}(U_\alpha)=\bigsqcup_\lambda U_\alpha^\lambda$, where each $p$ restricted over $U_\alpha^\lambda$ is a homeomorphism onto $U_\alpha$; and

(ii) if $U_\alpha^\lambda\cap U_\beta^\mu$ and $U_\alpha^\lambda\cap U_\beta^\nu$ are both nonempty, then $\mu=\nu$.

"Fool's covering spaces" are very close to overlays of R. H. Fox (see this paper in the first place and also this one), which I think are still better: they retain all nice properties of "fool's covering spaces" and have additional ones. An equivalent (see Lemma 7.3 in "Steenrod homotopy" or Mardesic-Matijevic) definition of an overlay is that it is

A third definition of overlays is by their monodromy: overlays are

functors $\check\Pi_1(X)\to Sets$, where $\check\Pi_1(X)$ is the Cech fundamental groupoid.

Note that $\check\Pi_1(X)=\Pi_1(X)$ if $X$ is locally connected and semi-locally simply-connected. For the equivalence, see Hernandez-Paricio (but note that his claim that Fox did his theory only for finite-sheeted overlays is not only incorrect but misleading; in fact, for finite-sheeted ones Fox shows that they reduce to coverings). In the connected case (more precisely, "Steenrod-connected" aka "pointed 1-movable" case), where the Cech fundamental groupoid reduces to the Cech fundamental group, the monodromy classification was essentially done by Fox, and a proof much shorter than either Fox's or Hernandez's can be found in "Steenrod homotopy" (see Corollary 7.5).

"Fool's covering spaces" are very close to overlays of R. H. Fox (see this paper in the first place and also this one), which I think are still better: they retain all nice properties of "fool's covering spaces" and have additional ones. An equivalent (see   "Steenrod homotopy", Lemma 7.3 or Mardesic-Matijevic) definition of an overlay is that it is

A third definition of overlays is by their monodromy. $d$-Sheeted overlays over a connected base $X$ (possibly $d=\infty$) are identified with

the homotopy set $[X,BS_d]$.

This is essentially the monodromy classification theorem of Fox; for a shorter proof and the above formulation see "Steenrod homotopy", Theorem 7.4. Another reformulation: overlays are

functors $pro$-$\Pi_1(X)\to Sets$, where $pro$-$\Pi_1$ is the fundamental pro-groupoid.

This is due to Hernandez-Paricio (but note that his claim that Fox did his theory only for finite-sheeted overlays is not only incorrect but misleading; in fact, for finite-sheeted ones Fox shows that they reduce to coverings). I'm not fully happy with the pro-groupoid definition because a pro-groupid is a whole diagram of groupoids. I would prefer something like "overlays are functors $\Pi_1\to Sets$, where $\Pi_1$ is the topologized Steenrod fundamental groupoid (which combines Steenrod $\pi_0$ and Steenrod $\pi_1$)" Such formulation is possible, at least, in a special case (see Corollary 7.5. in "Steenrod homotopy"). Over a base that is compact and Steenrod-connected (aka "pointed 1-movable"; in particular, this includes compact spaces that are connected and locally connected), overlays are identified with functors $\check\pi_1(X)\to Sets$, where $\check\pi_1$ is the topologized Cech (or Steenrod) fundamental group. Note that $\check\pi_1(X)=\pi_1(X)$ if $X$ is locally connected and semi-locally simply-connected.