The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is that one takes the 2-category of internal groupoids and localises with respect to internal functors $f:X\to Y$ such that $$X_1 \simeq X_0^2\times_{f_0^2,Y_0^2,(s,t)} Y_1$$ ("$f$ is fully faithful") and the composite map $$X_0\times_{f_0,Y_0,s}Iso(Y_1) \stackrel{pr_2}{\to} Iso(Y_1) \stackrel{t}{\to} Y_0 \qquad (1)$$ is some sort of "surjective" map. What this means depends on the site one works with. For example, one could be working with Lie groupoids, then (1) is a surjective submersion. Or topological groupoids and (1) admits local sections. The oldest reference I know to this sort of functor is in

M. Bunge and R. Par\'e, Stacks and equivalence of indexed categories, Cahiers Topologie Geom. Differentielle 20 (1979), no. 4, 373–399.

where they take $S$ to be finitely complete and regular, and (1) to be a regular epimorphism. Bunge and Par\'e call the externalisation of such an internal functor a weak equivalence.

Is this the earliest reference to this idea of weak equivalences of internal categories/groupoids?

  • $\begingroup$ Such a morphism makes sense for internal groupoids in any topos, where you replace "surjective submersion" with epimorphism. Then two such sheaves of groupoids are "weakly equivalent" if and only if their stackifications agree (as stacks over the topos, e.g., take the topos to be the topos of sheaves on some site). This is in Duskin's "An Outline of Non-Abelian Cohomology in a Topos", which is an '82 paper, however, I do not know if this appears earlier somewhere or not. $\endgroup$ – David Carchedi Sep 30 '10 at 5:46
  • $\begingroup$ Duskin's definition is subsumed by Bunge and Pare's, which is further subsumed by Everaert-Kieboom-van der Linden (this latter is post 2000, though), where EKvdL replace the class of regular epis by any class of maps, but for useful applications one takes this to be something a bit less general. $\endgroup$ – David Roberts Sep 30 '10 at 6:23

The stacks studied in the reference [link text][1] are fibrations over a base topos S not internal to a site S. The notion of an intrinsic stack originates with Lawvere, where one replace a site S with the site consisting of the class epimorphisms in the topos S. In this sense, the reference given is the first one in the literature. It is not needed that S be a topos as mentioned above by David Roberts.

[1]: Marta Bunge and Robert Pare, Stacks and Equivalence of Indexed Categories, Cahiers de Topologie et Geometrie Differentielle Categoriques XX-4 (1979) 373-399

  • $\begingroup$ Thanks Marta. I suspected as such, but wasn't sure if your definition was inspired by an earlier paper. $\endgroup$ – David Roberts Sep 8 '12 at 22:39
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    $\begingroup$ Welcome to MathOverflow, by the way! $\endgroup$ – David Roberts Sep 8 '12 at 22:40

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