If we fix a finite group $G$, there are two different useful homotopy theories on the set of $G$-equivariant topological spaces (which are CW complexes, say). One, the "weak" homotopy theory, is given by inverting all maps of spaces $X\to X'$ which are homotopy equivalences, and it produces a homotopy category equivalent to the category of spaces fibered over $BG$ (taking each equivariant space $X$ to $(X\times EG)/G$). The second is given by inverting all maps which are invertible up to equivariant homotopy, and it gives a new homotopy theory, namely the "strict" $G$-equivariant homotopy theory which is the unstable version of equivariant homotopy theory.
Now in algebraic geometry (as well as in the theory of smooth manifolds), given a finite group $G$ acting on a space $X$, there is a notion of orbifold (more generally, stack) $X/G$ which is an object with equivariant homotopy-theoretic flavor (in that it keeps track of stabilizers), except that the category of orbifolds allows the group $G$ to vary (and involves some glueing).
The topological space underlying any algebraic orbifold $X$ (so long as it is connected and suitably pointed) has a unique universal cover $\tilde{X}$, which is a simply connected space with no orbifold structure, and the "fundamental group" $\pi_1(X)$ acts on $\tilde{X},$ in a way that makes it reasonable to write $$X = \tilde{X}/G.$$
With this in mind, let's define a (pointed, connected) topological orbifold to be a pair $(\tilde{X}, G)$ written as $X = \tilde{X}/G$ with $G$ a discrete group acting on a (nice, e.g. CW) simply connected topological space $X$ with closed orbits and finite stabilizers. A map of orbifolds is a map of pairs $(\tilde{X}, G)\to (\tilde{X}', G')$ intertwining the actions in an evident way. Now there are once again two natural notions of homotopy equivalence.
- ("weak" homotopy theory) A map $\tilde{X}/G\to \tilde{X}'/G'$ is a homotopy equivalence either if $X' = \tilde{X}/H$ with $H\subset G$ a normal subgroup acting freely on $X$, and the map $G\to G'$ is surjective with kernel $H$, or if the map $G\to G'$ is an isomorphism and $\tilde{X}\to \tilde{X}'$ is a homotopy equivalence.
- ("strict" homotopy theory) A map $\tilde{X}/G\to \tilde{X}'/G'$ is a homotopy equivalence either if $X' = X/H$ with $H\subset G$ a normal subgroup acting freely on $X$ and $G\to G'$ is a surjection with kernel $H$, or if the map $G\to G'$ is an isomorphism and the map $\tilde{X}\to \tilde{X}'$ is a homotopy equivalence in the strict $G$-equivariant category.
The first notion of homotopy equivalence is treated in answers to this question. Here once again one can replace $\tilde{X}/G$ by the space $(\tilde{X}\times EG)/G$, and as far as I can tell, the homotopy category will boil down to something equivalent to the homotopy theory of topological spaces. I'm curious about the second notion of homotopy equivalence of topological orbifolds. Namely, is it a reasonable thing to study? Does it have model category structure? Is there a more general point of view (not involving universal covers) that allows dealing with more general stacks?
