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 Topos of toposes) is locally connected.

Definition

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

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

2. For every $A\in E$, the functor $(f/A)^* \colon E/A \to F/f^*A$ is cartesian closed.

3. $f^*$ commutes with dependent products – For any morphism $h\colon A\to B$ in $E$, the canonically defined natural transformation $(f/B)^* \circ \Pi_h \to \Pi_{f^*h} \circ (f/A)^*$ 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 $Set$

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

This is because the required Frobenius reciprocity condition

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

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

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

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

Strong adjunctions

The pair of adjoint functors $(f_! \dashv f^*)$ 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 a natural isomorphism

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

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

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

is a bijection. By adjunction this is the same as

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

Again by Yoneda, this is a bijection precisely if

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

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

Coreflectivity

Locally connected toposes are coreflective in Topos. See (Funk (1999)).

Other characterizations

• Let $(\mathcal{C}, J)$ be a site and $S$ be a sieve on the object $U$. $S$ is called connected when $S$ viewed as a full subcategory of $\mathcal{C}/U$ is connected. The site is called locally connected if every sieve is connected. For a bounded geometric morphism $p:\mathcal{E}\to\mathcal{S}$ the following holds: $p$ is locally connected iff there exists a locally connected internal site in $\mathcal{S}$ such that $\mathcal{E}\simeq Sh(\mathcal{C},J)$. (cf. Johnstone (2002), pp.656-658)

• Caramello (2012) gives syntactic characterizations of geometric theories whose classifying topos is locally connected.

The same paper also contains the following characterization:

• A Grothendieck topos is locally connected iff it has a separating set of (coproduct) indecomposable objects.

Variations in the context of the Nullstellensatz

Johnstone (2011) studies several subclasses of locally connected geometric morphisms in the context of Lawvere‘s theory of cohesion and the Nullstellensatz. He calls a locally connected morphism $p$ stably locally connected if $p_!$ preserves finite products. According to the above remark this implies that $p$ is connected. Slightly stronger is the preservation of all finite limits by $p_!$: these $p$ are called totally connected geometric morphisms.

Examples

• If the terminal global section geometric morphism $E \to Set$ 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 $j\colon Sh(U)\to Sh(X)$. Then any $A\in Sh(X)$ can be identified with a space (or locale) $A$ equipped with a local homeomorphism $A\to X$, in such a way that $Sh(X)/A \simeq Sh(A)$. Moreover, $j^*A \in Sh(U)$ can be identified with the pullback of $A\to X$ along $U$, and so $Sh(U)/j^*A \simeq Sh(j^*A)$ 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 $(j/A)^*\colon Sh(X)/A \to Sh(U)/j^*A$ is cartesian closed for any $A$. Hence $j$ is locally connected.

Related entries

• connected geometric morphism

• local geometric morphism

• totally connected geometric morphism

References

The case of $Sh(X)$ for a topological space $X$ was an exercise (p.417) in SGA4:

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), Springer LNM vol.269 (1972).

The concept relative to other bases was introduced in the following paper:

• Michael Barr, Robert Paré, Molecular Toposes , JPAA 17 (1980) pp.127-152.

The standard reference is section C3.3 of

• Peter Johnstone, Sketches of an Elephant , Oxford UP 2002.

Further references include

• Olivia Caramello, Syntactic Characterizations of Properties of Classifying Toposes , TAC 26 no.6 (2012) pp.176-193. (pdf)

• Jonathon Funk, The locally connected coclosure of a Grothendieck topos, JPAA 137 (1999) pp.17-27.

• Peter Johnstone, Remarks on Punctual Local Connectedness , TAC 25 no.3 (2011) pp.51-63. (pdf)

• Ieke Moerdijk, Continuous fibrations and inverse limits of toposes , Comp. Math. 58 (1986) pp.45-72. (pdf)

• Ieke Moerdijk, Gavin Wraith, Connected and locally connected toposes are path connected , Trans. AMS 295 (1986) pp.849-859. (pdf)