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 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 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$ , viewed as a space fibered over $ BG$ . 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?