nLab
locally connected geometric morphism

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

A geometric morphism is locally connected if it behaves as though its fibers are locally connected spaces. In particular, a Grothendieck topos EE is locally connected iff the unique geometric morphism to Set (the terminal Grothendieck topos, i.e. the point in the category Toposes of toposes) is locally connected.

Definition

A geometric morphism (f *f *):Ff *f *E (f^* \dashv f_*) : F \underoverset{f_*}{f^*}{\leftrightarrows} E is locally connected if it satisfies the following equivalent conditions:

  1. It is essential, i.e. f *f^* has a left adjoint f !f_!, and moreover f !f_! can be made into an EE-indexed functor.

  2. For every AEA\in E, the functor f *:E/AF/f *Af^* \colon E/A \to F/f^*A is cartesian closed.

  3. f *f^* commutes with dependent products – For any morphism h:ABh\colon A\to B in EE, the canonically defined natural transformation f *Π hΠ f *hf *f^* \circ \Pi_h \to \Pi_{f^*h} \circ f^* is an isomorphism.

Properties

Relation to connectedness

If ff is locally connected, then it makes sense to think of the left adjoint f !f_! as assigning to an object of FF its “set of connected components” in EE. In particular, if ff is locally connected, then it is moreover connected if and only if f !f_! preserves the terminal object. However, not every connected geometric morphism is locally connected.

Over SetSet

Over the base topos E=E = Set every connected topos which is essential is automatically locally connected.

This is because the required Frobenius reciprocity condition

f !(A×f *(B))f !(A)×B f_!(A \times f^* (B)) \simeq f_!(A) \times B

is automatically satisfied, using that cartesian product with a set is equivalently a coproduct

A×B= aAB, A \times B = \coprod_{a \in A} B \,,

that the left adjoint f !f_! preserves coproducts, and that for f *f^* full and faithful we have f !f *Idf_! f^* \simeq Id.

Strong adjunctions

The pair of adjoint functors (f !f *)(f_! \dashv f^*) in a locally connected geometric morphisms forms a “strong adjunction” in that it holds also for the internal homs in the sense that there is an natural isomorphism

[f !(X),A]f *[X,f *A] [f_!(X), A] \simeq f_* [X, f^* A]

for all X,AX, A. This follows by duality from the Frobenius reciprocity that characterizes f *f_* as being a cartesian closed functor:

by the Yoneda lemma, the morphism in question is an isomorphism if for all objects A,B,XA,B, X the morphism

Hom(X,[f !(A),B])Hom(X,f *[A,f *(B)]) Hom(X, [f_!(A), B]) \stackrel{}{\to} Hom(X,f_*[A,f^*(B)])

is a bijection. By adjunction this is the same as

Hom(X×f !(A),B)Hom(f !(f *(X)×A),B). Hom(X \times f_!(A), B) \stackrel{\simeq}{\to} Hom(f_!(f^*(X) \times A), B) \,.

Again by Yoneda, this is a bijection precisely if

f !(f *(X)×A)X×f !(A) f_!(f^*(X) \times A) \to X \times f_!(A)

is an isomorphism. But this is the Frobenius reciprocity condition on f *f^*.

Coreflectivity

Locally connected toposes are coreflective in Topos. See (Funk).

Examples

  • If the terminal global section geometric morphism ESetE \to Set is locally connected, one calls EE a locally connected topos. More generally, if ESE\to S is locally connected, we may call EE a locally connected SS-topos.

  • Let XX be a topological space (or a locale) and UXU\subseteq X an open subset, with corresponding geometric embedding j:Sh(U)Sh(X)j\colon Sh(U)\to Sh(X). Then any ASh(X)A\in Sh(X) can be identified with a space (or locale) AA equipped with a local homeomorphism AXA\to X, in such a way that Sh(X)/ASh(A)Sh(X)/A \simeq Sh(A). Moreover, j *ASh(U)j^*A \in Sh(U) can be identified with the pullback of AXA\to X along UU, and so Sh(U)/j *ASh(j *A)Sh(U)/j^*A \simeq Sh(j^*A) similarly. Noting that j *AAj^*A \to A is again the inclusion of an open subset, and using the fact that the inverse image part of any open geometric embedding is cartesian closed, we see that (j/A) *:Sh(X)/ASh(U)/j *A(j/A)^*\colon Sh(X)/A \to Sh(U)/j^*A is cartesian closed for any AA. Hence jj is locally connected.

References

The canonical textbook reference is section C3.3 of

Further references include

  • J. Funk, The locally connected coclosure of a Grothendieck topos, Journal of Pure and Applied Algebra Volume 137, Issue 1, 7 (1999), Pages 17-27

  • Ieke Moerdijk, Gavin Wraith, Connected and locally connected toposes are path connected (JSTOR)

Revised on August 31, 2014 21:51:20 by David Roberts (129.127.211.139)