Connected locally connected toposes are path-connected

Moerdijk, I.; Wraith, G. C.

doi:10.1090/S0002-9947-1986-0833712-3

Connected locally connected toposes are path-connected
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by I. Moerdijk and G. C. Wraith PDF

Trans. Amer. Math. Soc. 295 (1986), 849-859 Request permission

Abstract:

A conjecture of A. Joyal is proved, which states that, in contrast to topological spaces, toposes which are connected and locally connected are also path-connected. The reason for this phenomenon is the triviality of cardinality considerations in the topos-theoretic setting; any inhabited object pulls back to an enumerable object under some open surjective geometric morphism. This result points towards a homotopy theory for toposes.

References

Michael Barr and Robert Paré, Molecular toposes, J. Pure Appl. Algebra 17 (1980), no. 2, 127–152. MR 567064, DOI 10.1016/0022-4049(80)90080-8
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
M. P. Fourman and R. J. Grayson, Formal spaces, The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) Stud. Logic Found. Math., vol. 110, North-Holland, Amsterdam, 1982, pp. 107–122. MR 717242, DOI 10.1016/S0049-237X(09)70126-0
Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354653

Function spaces in the category of locales

John R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 5–32. MR 358725, DOI 10.7146/math.scand.a-11409
Peter T. Johnstone, Open maps of toposes, Manuscripta Math. 31 (1980), no. 1-3, 217–247. MR 576498, DOI 10.1007/BF01303275
Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR 698074
Peter T. Johnstone, How general is a generalized space?, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 77–111. MR 787824
Peter Johnstone and André Joyal, Continuous categories and exponentiable toposes, J. Pure Appl. Algebra 25 (1982), no. 3, 255–296. MR 666021, DOI 10.1016/0022-4049(82)90083-4
André Joyal and Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71. MR 756176, DOI 10.1090/memo/0309
Karl Menger, Über die Dimension von Punktmengen III, Monatsh. Math. Phys. 36 (1929), no. 1, 193–218 (German). Zur Begründung einer axiomatischen Theorie der Dimension. MR 1549681, DOI 10.1007/BF02307613
Ieke Moerdijk, Continuous fibrations and inverse limits of toposes, Compositio Math. 58 (1986), no. 1, 45–72. MR 834047

Path-lifting for Grothendieck toposes

R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR 507446