Abstract
The straight answer is no. Topoi are too set-like to occur as categories of sets with topological structure. However, if A is a category of sets with structure, and if A has enough substructures, then A has a full and dense embedding into a complete quasitopos of sets with structure. There is a minimal embedding of this type; it embeds e.g. topological spaces into the quasitopos of Choquet spaces. Quasitopoy are still very set-like. They are cartesian closed, and all colimits in a quasitopos are preserved by pullbacks. Thus quasitopoi are in a sense ultra-convenient categories for topologists. Quasitopoi inherit many properties from topoi. For example, the theory of geometric morphisms of topoi remains valid, almost without changes, for quasitopoi.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Antoine, P., Extension minimale de la catégorie des espaces topologiques. C. R. Acad. Sc. Paris 262 (1966), sér. A, 1389–1392.
Antoine, P., Etude élémentaire des catégories d'ensembles structurés. Bull. Soc. Math. Belgique 18 (1966), 142–166 and 387–414.
Birkhoff, G., Lattice Theory. Rev. Ed. Providence, 1948.
Choquet, G., Convergences. Ann. Univ. Grenoble, Sect. Sc. Math. Phys. (N.S.) 23 (1948), 57–112.
Day, B., A reflection theorem for closed categories. J. Pure Appl. Alg. 2 (1972), 1–11.
Day, B., Limit spaces and closed span categories. Category Seminar, Sidney 1972/73, pp. 65–74. Lecture Notes in Math. 420 (1974).
Day, B.J., and G.M. Kelly, On topological quotient maps. Proc. Camb. Phil. Soc. 67 (1970), 553–558.
Fischer, H.R., Limesräume. Math. Annalen 137 (1959), 269–303.
Freyd, P., Aspects of topoi. Bull. Austral. Math. Soc. 7 (1972), 1–76.
Herrlich, H., Cartesian closed topological categories. Preprint (1974).
Jurchescu, A., and M. Lascu, Morphisme stricte, categoriî cantoriene, functori de completare. Studiî Cerc. Mat. 18 (1966), 219–234.
Kelly, G.M., Monomorphisms, epimorphisms, and pullbacks. J. Austral. Math. Soc. 9 (1969), 124–142.
Kock, A., and G. C. Wraith, Elementary toposes. Lecture Note Series, no. 30. Matematisk Institut, Aarhus Universitet, 1971.
Kowalsky, H.-J., Limesräume und Komplettierung. Math. Nachr. 12 (1954), 301–340.
MacLane, S., Categories for the Working Mathematician. Springer, 1971.
Penon, J., Quasi-topos. C. R. Acad. Sc. Paris, 276 (1973), Sér. A, 237–240.
Schroder, M., Solid convergence spaces. Bull. Austral. Math. Soc. 8 (1973), 443–459.
Steenrod, N.E., A convenient category of topological spaces. Michigan Math. J. 14 (1967), 133–152.
Stout, L.N., General topology in an elementary topos. Ph.D. Thesis, University of Illinois, 1974.
Tierney, M., Axiomatic sheaf theory: some constructions and applications. C.I.M.E. III Ciclo 1971, Varenna 12–21 Settembee. Cremonese, Roma, 1973.
Verdier, J.L., SGA 4/I, Théorie des topos et cohomologie étale des schémas, Exposés I–IV. 2me. éd. Lecture Notes in Math. 269 (1972).
Wyler, O., On the categories of general topology and topological algebra. Arch. Math. 22 (1971), 7–17.
Wyler, O., Top categories and categorical topology. Gen. Top. Appl. 1 (1971), 17–28.
Wyler, O., Quotient maps. Gen. Top. Appl. 3 (1973), 149–160.
Wyler, O., Convenient categories for topology. Gen. Top. Appl. 3 (1973), 225–242.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1976 Springer-Verlag
About this paper
Cite this paper
Wyler, O. (1976). Are there topoi in topology?. In: Binz, E., Herrlich, H. (eds) Categorical Topology. Lecture Notes in Mathematics, vol 540. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080884
Download citation
DOI: https://doi.org/10.1007/BFb0080884
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07859-3
Online ISBN: 978-3-540-38118-1
eBook Packages: Springer Book Archive