nLab
topological site

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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

Definition

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

A topos \mathcal{E} is called topological when it is equivalent to a topos Sh(𝒞,J)Sh(\mathcal{C}, J) with (𝒞,J)(\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 𝒞\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 𝒞\mathcal{C} have objects the locally compact subspaces of some n\mathbb{R}^n, nn\in\mathbb{N} varying, and morphisms the C C^\infty-maps. The topos Sh(𝒞,J)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 AA be a T UT_U-propositional theory, and let A 0A_0 be the corresponding locale in SetSet (so Sh(A 0)Sh(A_0) classifies AA-models in Grothendieck toposes). Let Sh(𝒞,J)Sh(\mathcal{C}, J) be a topological topos. Then the object of AA-models in Sh(𝒞,J)Sh(\mathcal{C}, J) is given by the sheaf Cts(,A 0):𝒞 opSetCts(-, A_0) : \mathcal{C}^{op}\to Set which assigns to a space XX in 𝒞\mathcal{C} the set of all continuous maps from XX to A 0A_0.

This occurs as cor.1.5 in Moerdijk-Reyes (1984). Here T UT_U is the pendant for locales of the T 1T_1-separation property: a locale XX is T UT_U if for all locale maps f,g:YXf,g:Y\to X the relation fgf\leq g implies f=gf=g (cf. the Elephant II, p.501).

Proposition

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

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

References

Last revised on November 15, 2019 at 03:32:56. See the history of this page for a list of all contributions to it.