nLab topological site

Idea

A topological site is a site $(\mathcal{C}, J)$ that generalizes the site that underlies Giraud’s gros topos of topological spaces. The aim is to get in $Sh(\mathcal{C}, J)$ a well-behaved category of topological spaces.

Definition

A site $(\mathcal{C}, J)$ is called topological if $\mathcal{C}$ is a category of topological spaces and continuous maps, which is closed under open inclusions i.e. if $X\in\mathcal{C}$ and $U\in\mathcal{O}(X)$ then $U\hookrightarrow X$ is a morphism in $\mathcal{C}$ and $J$ is the open cover topology i.e. $J$ is generated by families $\{ U_i\hookrightarrow X\}$ where the $U_i$ are open and jointly cover $X$.

A topos $\mathcal{E}$ is called topological when it is equivalent to a topos $Sh(\mathcal{C}, J)$ with $(\mathcal{C}, J)$ a topological site.

Remark

This definition follows Moerdijk-Reyes (1984) but some variation is possible here e.g. one could additionally demand that $\mathcal{C}$ is closed under finite limits and contains the real numbers $\mathbb{R}$ (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).

Examples

• Let $\mathcal{C}$ have objects the locally compact subspaces of some $\mathbb{R}^n$, $n\in\mathbb{N}$ varying, and morphisms the $C^\infty$-maps. The topos $Sh(\mathcal{C}, J)$ is called the euclidean topos in Moerdijk-Reyes (1984) where its properties and in particular its relations with the Dubuc topos are explored.

Properties

Proposition

Let $A$ be a $T_U$-propositional theory, and let $A_0$ be the corresponding locale in $Set$ (so $Sh(A_0)$ classifies $A$-models in Grothendieck toposes). Let $Sh(\mathcal{C}, J)$ be a topological topos. Then the object of $A$-models in $Sh(\mathcal{C}, J)$ is given by the sheaf $Cts(-, A_0) : \mathcal{C}^{op}\to Set$ which assigns to a space $X$ in $\mathcal{C}$ the set of all continuous maps from $X$ to $A_0$.

This occurs as cor.1.5 in Moerdijk-Reyes (1984). Here $T_U$ is the pendant for locales of the $T_1$-separation property: a locale $X$ is $T_U$ if for all locale maps $f,g:Y\to X$ the relation $f\leq g$ implies $f=g$ (cf. the Elephant II, p.501).

Proposition

Let $Sh(\mathcal{C}, J)$ be a topological topos. Then the Fan theorem holds in $Sh(\mathcal{C}, J)$. If furthermore all $X\in\mathcal{C}$ are locally compact then Bar induction? holds as well.

This is contained in props.1.7-8 in Moerdijk-Reyes (1984).

Related entries

• Johnstone's topological topos

• open cover

• continuous truth

• gros topos

References

• 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$_1$ 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