Moerdijk, I.; Wraith, G. C. Connected locally connected toposes are path-connected. (English) Zbl 0592.18003 Trans. Am. Math. Soc. 295, 849-859 (1986). This paper proves the result stated in its title (and a bit more besides: they are also semi-locally path-connected). At first sight, it is a surprising result, since the corresponding statement for spaces is well known to be false; the key to understanding how it can be true in the more general context of toposes lies in a reappraisal of the notion of path-connectedness. One says that a topos \({\mathcal E}\) in path-connected if the canonical map ”evaluation at endpoints” P\({\mathcal E}\to {\mathcal E}\times {\mathcal E}\) is surjective, where P\({\mathcal E}\) is the topos of paths in \({\mathcal E}\) (i.e. the exponential \({\mathcal E}^{{\mathcal I}}\), where \({\mathcal I}\) is (the topos of sheaves on) the unit interval). In the category of spaces, this is of course equivalent to the usual definition; the crucial point is that topos exponentials are in general ”larger” than spatial ones. The proof of the result is a nice application of the topos-theoretic technique of changing base, so as to reduce the problem to successively simpler cases: first to the case when \({\mathcal E}\) is localic, and then to the case when it is countably presented (in which case the classical Menger-Moore proof, for complete metric spaces, can easily be mimicked). Reviewer: P.Johnstone Cited in 2 ReviewsCited in 11 Documents MSC: 18B25 Topoi 54D05 Connected and locally connected spaces (general aspects) Keywords:connected locally connected toposes; semi-locally path-connected; topos of paths; topos of sheaves PDFBibTeX XMLCite \textit{I. Moerdijk} and \textit{G. C. Wraith}, Trans. Am. Math. Soc. 295, 849--859 (1986; Zbl 0592.18003) Full Text: DOI References: [1] Michael Barr and Robert Paré, Molecular toposes, J. Pure Appl. 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