nLab
locally connected geometric morphism

Idea

A geometric morphism is locally connected if it behaves as though its fibers are locally connected spaces. In particular, a Grothendieck topos $E$ 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 $(f^{*}⊣f_{*}):F\underset{f_{*}}{\overset{f^{*}}{\leftrightarrows }}E$ is locally connected if it satisfies the following equivalent conditions:

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

2. For every $A\in E$, the functor $f^{*}:E/A\to F/f^{*}A$ is cartesian closed.

3. $f^{*}$ commutes with dependent products – For any morphism $h:A\to B$ in $E$, the canonically defined natural transformation $f^{*}\circ {\Pi }_h\to {\Pi }_{f^{*}h}\circ f^{*}$ is an isomorphism.

Properties

Relation to connectedness

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

Over $\mathrm{Set}$

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

This is because the required Frobenius reciprocity condition

$$f_{!}(A\times f^{*}(B))\simeq f_{!}(A)\times B$$f_!(A \times f^* (B)) \simeq f_!(A) \times B

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

$$A\times B=\coprod _{a\in A}B,$$A \times B = \coprod_{a \in A} B \,,

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

Strong adjunctions

The pair of adjoint functors $(f_{!}⊣f^{*})$ 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

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

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

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

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

is a bijection. By adjunction this is the same as

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

$$f_{!}(f^{*}(X)\times A)\to X\times f_{!}(A)$$f_!(f^*(X) \times A) \to X \times f_!(A)

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

Coreflectivity

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

Examples

• If the terminal global section geometric morphism $E\to \mathrm{Set}$ is locally connected, one calls $E$ a locally connected topos. More generally, if $E\to S$ is locally connected, we may call $E$ a locally connected $S$-topos.

• Let $X$ be a topological space (or a locale) and $U\subseteq X$ an open subset, with corresponding geometric embedding $j:\mathrm{Sh}(U)\to \mathrm{Sh}(X)$. Then any $A\in \mathrm{Sh}(X)$ can be identified with a space (or locale) $A$ equipped with a local homeomorphism $A\to X$, in such a way that $\mathrm{Sh}(X)/A\simeq \mathrm{Sh}(A)$. Moreever, $j^{*}A\in \mathrm{Sh}(U)$ can be identified with the pullback of $A\to X$ along $U$, and so $\mathrm{Sh}(U)/j^{*}A\simeq \mathrm{Sh}(j^{*}A)$ similarly. Noting that $j^{*}A\to A$ 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 $(j/A{)}^{*}:\mathrm{Sh}(X)/A\to \mathrm{Sh}(U)/j^{*}A$ is cartesian closed for any $A$. Hence $j$ is locally connected.

References

The canonical textbook reference is section C3.3 of

• Peter Johnstone, Sketches of an Elephant

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, Connectedd and locally connected toposes are path connected (JSTOR)