TITLE: Who first came up with the idea of essential/Morita equivalence of internal groupoids/categories?
QUESTION [8 upvotes]: 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. Paré, 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 Paré 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?

REPLY [11 votes]: The stacks studied in the reference [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