van Kampen theorems for toposes (Q1413097)

\textit{R. Brown} and \textit{G. Janelidze} [J. Pure Appl. Algebra 119, 255-263 (1997; Zbl 0882.18005)] proved a `van Kampen theorem' for lextensive categories: in this context the theorem becomes the assertion that, under suitable hypotheses, pseudofunctors defined on the duals of such categories map certain pushouts to pullbacks. The authors of the present paper first introduce a notion of `extensive \(2\)-category' to which the Brown-Janelidze result can be extended. Then they show that the \(2\)-categories of toposes over a base, and of locally connected toposes over a base, satisfy their definition of extensiveness, which enables them to recover the `classical' van Kampen theorem for the fundamental groupoids of toposes, defined in terms of coverings. \(\{\)There is a minor error in Proposition 2.4: in the statement of this Proposition, the words `fully faithful' should be replaced by `pseudomonic'. However, this does not affect any of the other results in the paper\(\}\).