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.