Tuesday, 29 October 2002
2:30 - 4:00   Marta Bunge
Stack completions revisited
Abstract: The notion of a stack over a topos S has been formulated first in terms of a site (Grothendieck-Giraud 1972) and then in terms of the canonical topology of regular epis (Bunge-Pare 1979). Although the latter offers a simpler approach and is meaningful for arbitrary elementary toposes, it is tied up with the question of the representability of stack completions for the category objects (``axiom of stack completions'').
The stack completion (for the regular epis) of a (localic) groupoid G in a topos S is the S-indexed category Tors^1(G) of G-torsors. An alternative description, for an etale complete (localic) groupoid G, has been shown [Bunge 1979, 1990] to be describable in terms of the groupoid of (essential) points of the classifying topos BG (``classification theorem'') .
In this lecture I will, after discussing briefly the above, (1) first apply the above in order to justify (a refinement of) the assertion made in [Bunge-Moerdijk 1997, after Bunge 1992] that the fundamental groupoid of a locally connected topos E over S is (weakly equivalent to) a prodiscrete groupoid in S, and (2) then discuss a 2-categorical version of the above ``classification theorem'' involving 2-descent, 2-stacks and 2-torsors, and which attempts to explain the role of gerbes, liens and bouquets (Grothendieck, Giraud, Duskin, Breen, Street, Mauri-Tierney) in this context, while pointing to higher-dimensional analogues.