Cohomology and homotopy
In higher category theory
A geometric morphism is locally connected if it behaves as though its fibers are locally connected spaces. In particular, a Grothendieck topos 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.
A geometric morphism is locally connected if it satisfies the following equivalent conditions:
It is essential, i.e. has a left adjoint , and moreover can be made into an -indexed functor.
For every , the functor is cartesian closed.
commutes with dependent products – For any morphism in , the canonically defined natural transformation is an isomorphism.
Relation to connectedness
If is locally connected, then it makes sense to think of the left adjoint as assigning to an object of its “set of connected components” in . In particular, if is locally connected, then it is moreover connected if and only if preserves the terminal object. However, not every connected geometric morphism is locally connected.
Over the base topos Set every connected topos which is essential is automatically locally connected.
This is because the required Frobenius reciprocity condition
is automatically satisfied, using that cartesian product with a set is equivalently a coproduct
that the left adjoint preserves coproducts, and that for full and faithful we have .
The pair of adjoint functors 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
for all . This follows by duality from the Frobenius reciprocity that characterizes as being a cartesian closed functor:
by the Yoneda lemma, the morphism in question is an isomorphism if for all objects the morphism
is a bijection. By adjunction this is the same as
Again by Yoneda, this is a bijection precisely if
is an isomorphism. But this is the Frobenius reciprocity condition on .
Locally connected toposes are coreflective in Topos. See (Funk (1999)).
Let be a site and be a sieve on the object . is called connected when viewed as a full subcategory of is connected. The site is called locally connected if every sieve is connected. For a bounded geometric morphism the following holds: is locally connected iff there exists a locally connected internal site in such that . (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 stably locally connected if preserves finite products. According to the above remark this implies that is connected. Slightly stronger is the preservation of all finite limits by : these are called totally connected geometric morphisms.
If the terminal global section geometric morphism is locally connected, one calls a locally connected topos. More generally, if is locally connected, we may call a locally connected -topos.
Let be a topological space (or a locale) and an open subset, with corresponding geometric embedding . Then any can be identified with a space (or locale) equipped with a local homeomorphism , in such a way that . Moreover, can be identified with the pullback of along , and so similarly. Noting that 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 is cartesian closed for any . Hence is locally connected.
The case of for a topological space was an exercise (p.417) in SGA4:
The concept relative to other bases was introduced in the following paper:
The standard reference is section C3.3 of
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)