A topological site is a site that generalizes the site that underlies Giraud’s gros topos of topological spaces. The aim is to get in a well-behaved category of topological spaces.
A site is called topological if is a category of topological spaces and continuous maps, which is closed under open inclusions i.e. if and then is a morphism in and is the open cover topology i.e. is generated by families where the are open and jointly cover .
A topos is called topological when it is equivalent to a topos with a topological site.
This definition follows Moerdijk-Reyes (1984) but some variation is possible here e.g. one could additionally demand that is closed under finite limits and contains the real numbers (cf. Mac Lane-Moerdijk 1994). Instead of topological spaces one could use locales - this is pursued by Fourman (1983,1984,2013). Compare also the approach taken in Montañez (2013).
Let be a -propositional theory, and let be the corresponding locale in (so classifies -models in Grothendieck toposes). Let be a topological topos. Then the object of -models in is given by the sheaf which assigns to a space in the set of all continuous maps from to .
This occurs as cor.1.5 in Moerdijk-Reyes (1984). Here is the pendant for locales of the -separation property: a locale is if for all locale maps the relation implies (cf. the Elephant II, p.501).
Let be a topological topos. Then the Fan theorem holds in . If furthermore all are locally compact then Bar induction? holds as well.
This is contained in props.1.7-8 in Moerdijk-Reyes (1984).
M.Artin, A.Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4 , LNM 269 Springer Heidelberg 1972.
Michael Fourman, T Spaces over Topological Sites , JPAA 27 (1983) pp.223-224. link
Michael Fourman, Continuous Truth I: non-constructive objects , pp.161-180 in Lolli, Longo, Marcja (eds.), Proc. Logic Colloquium, Florence 1982 , Elsevier Amsterdam 1984. (draft)
Michael Fourman, Continuous Truth II: reflections , pp,153-167 in LNCS 8071 Springer Heidelberg 2013. (draft)
Gerrit Van Der Hoeven, Ieke Moerdijk, Sheaf models for choice sequences , APAL 27 (1984) pp.63-107.
Peter Johnstone, On a topological topos , Proc. London Math. Soc. (3) 38 (1979) pp.237–271.
Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (pp.113, 325ff, 416)
Ieke Moerdijk, Gonzalo E. Reyes, Smooth Spaces versus Continuous Spaces in Models of Synthetic Differential Geometry , JPAA 32 (1984) pp.143-176. link
R. Montañez, Topoi generated by topological spaces , Talk CT15 Aveiro 2015. (pdf-slides)
Example A2.1.11 in the Elephant
Last revised on July 5, 2021 at 11:54:17. See the history of this page for a list of all contributions to it.