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I'm teaching an elementary class about fundamental groups and covering spaces. It was very useful to use "fool's covering spaces" of a space $X$, defined as functors $\Pi_1(X)\to Sets$, where $\Pi_1(X)$ is the fundamental groupoids of $X$. In a more "covering space way", a fool's covering space can be described as a set $Y$, a map $p:Y\to X$, and a map $p^{-1}(x_1)\to p^{-1}(x_2)$ for any path between $x_1, x_2\in X$, satisfying the obvious properties.

Is there a standard name for "fool's covering spaces"? Calling them "functors $\Pi_1(X)\to Sets$ " is a bit heavy for the class.

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What exactly is the covering space among your "fool's covering spaces"? I'm confused as to why you're talking about sets rather than spaces and maps. –  Ryan Budney Dec 15 '10 at 13:36
    
@Ryan: If $p: \hat{X} \to X$ is the universal covering space, then $\Pi_1(X)$ acts on $p$ in the category of etale spaces over $X$, and given a functor $F: \Pi_1(X) \to Set$, the corresponding covering space is obtained as the tensor product $p \otimes_{\Pi_1(X)} F$. @Trial: why "fool's"? If we relax the usual surjectivity condition of covering spaces and allow empty fibers, isn't the category of covering spaces over $X$ (for nice $X$) equivalent to the toppos of functors $\Pi_1(X) \to Set$? –  Todd Trimble Dec 15 '10 at 14:08
    
Sorry for being unclear. Any covering space is also a "fool's covering space". For a locally path-connected and semilocally 1-connected spaces the two notions are equivalent (this is considered a "difficult theorem" in the cours). In a fool's covering space the set $Y$ is just a set, with no topology (if it's uclear, take "functor $\Pi_1(X)\to Sets$ " as a definition of fool's covering space, and forget the other description). –  Pavol S. Dec 15 '10 at 14:15
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I think it would be misleading to mistake the functor you're talking about with the associated covering space. Generally I'm not aware of a standard name for this process. I suppose I'd call it the monodromy classification of covering spaces or the action of $\pi_1$ on the fibre, or something like that -- I don't think this categorical perspective is much more than a "repackaging" of a classical theorem, so I just call these things by their classical names. –  Ryan Budney Dec 15 '10 at 14:47
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@Ryan: here are 2 pedagogical reasons for giving "fool' covering spaces" independent life and name: 1. the fact that they are equivalent to ordinary covering spaces (if the base is nice) can be described by lifting the topology from $X$ to $Y$, which is an easy operation. 2. Using the classification of covering spaces (for nice base), van Kampen theorem (for nice spaces) is the trivial statement that if something is locally (in $X$) a covering space then it is a covering space. Without this classification and for arbitrary spaces it is the same locality statement for fool's covering spaces. –  Pavol S. Dec 15 '10 at 16:13
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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

a covering that is induced from some covering over a polyhedron (or equivalently from some covering over a locally connected semi-locally simply-connected space).

Fox's original (equivalent) definition 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 $\mu=\nu$.

Condition (i) of course amounts to a definition of a covering in the usual sense.

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.

Finally, I should mention that over a compact (metric) base, overlays can also be defined (Theorem 7.6 in "Steenrod homotopy") as

coverings in the category of uniform spaces.

Such uniform coverings have been studied by I. M. James in his book "Introduction to Uniform spaces"; see Brodsky-Dydak-Labuz-Mitra for a clarification of James' definition (the latter paper also has some relevant followups). This is really saying that overlays are precisely those coverings for which a metric on the base can be "lifted" to a metric in the total space. (Note that the compact base has a unique uniformity: as everyone might remember, every continuous function on a compact space is uniformly continuous.)

DISCLAIMER: Following Fox, I have been assuming all spaces to be metrizable :) It is known that this is not a real restriction, and everything extends to arbitrary spaces, perhaps with minor modifications (see Mardesic-Matijevic's paper, which also has many additional references about overlays; also the papers by Dydak-et-al. and Hernandez-Paricio may be relevant to this point) However, I prefer being ignorant of the non-metrizable world and so don't follow these modifications or whether they are needed.

SUMMARY: For purposes of proving something about coverings of locally connected semi-locally simply-connected spaces usual covering work fine. For purposes of proving anything in topology beyond these restrictions, you would definitely need overlays, rather than "fool's covering spaces". But admittedly overlays are slightly harder to define. Thus for purposes of defining a formal concept which agrees with coverings for "nice" spaces and is not intended to be used for proving anything beyond "nice" spaces, "fool's covering spaces" suit well; I would call them e.g. path-overlays.

By the way, I like the idea about the Seifert-van Kampen theorem; I think if combined with overlays, it should give a Seifert-van Kampen theorem in Steenrod homotopy, which would be an interesting result.

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I asked a terminology question, and instead I learned something very interesting. Thanks a lot! –  Pavol S. Dec 15 '10 at 22:56
    
Sorry, what I first attributed to Hernandez-Paricio was not exactly what he proved. –  Sergey Melikhov Dec 16 '10 at 1:02
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In the Algebraic Topology literature, what you describe would be called a local system of sets on $X$. In general a local system on a space $X$ is a covariant functor from $\Pi_1(X)$ to some category.

In Steenrod's definition of homology with local coefficients, he uses a local system of abelian groups as coefficients (ordinary homology corresponds to the constant functor). This is explained nicely in G.W Whitehead's book Elements of Homotopy Theory.

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This is perhaps not exactly what I wanted: a local system should be a functor from the Cech fundamental groupoid, while I was asking about the usual (path) fundamental groupoid (e.g. the application to Van Kampen theorem was meant for the usual (path) fundamental group). –  Pavol S. Jan 3 '11 at 20:53
    
@Trial-well, I sabotaged my own answer by link I chose, which discusses the more modern, algebro-geometric usage of the term. But the original topological definition uses the usual groupoid (see the References section half way down the linked page). –  Mark Grant Jan 3 '11 at 22:02
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