A local geometric morphism between toposes is
When we regard as a topos over , so that is regarded as its global section geometric morphism in the category of toposes over , then we say that is a local -topos. In this case we may label the functors involved as
to indicate that if we think of as sending a space to its underlying -object of points by forgetting cohesion, then creates the discrete space/discrete object and the codiscrete space/codiscrete object on an object in .
(As just stated, it is automatic in the case over that this is furthermore a full and faithful functor.)
Another way of stating this is that a Grothendieck topos is local if and only if the terminal object is connected and projective (since this means precisely that preserves colimits, and therefore has a right adjoint by virtue of an adjoint functor theorem). Another term for this: we say is tiny (atomic). Notice the similarity to the concept of amazing right adjoint (the difference being that this is a right adjoint not to the external but to the internal hom out of 1.)
Note also that in an infinitary-extensive category an object is connected as soon as preserves binary coproducts (see connected object). Moreover, a coproduct-preserving functor between toposes preserves coequalizers as soon as it preserves epimorphisms, since any coequalizer can be constructed as the quotient of an equivalence relation generated using images and countable coproducts and quotients of equivalence relations are effective. Thus, we can say that a Grothendieck topos is local iff preserves binary coproducts and epis. Moreover, a cospan in a topos is a coproduct diagram iff and are disjoint monos whose union is all of ; thus preserves binary coproducts as soon as it preserves the initial object and binary unions. This leads to the following equivalent form of “locality” that makes sense even for elementary toposes:
An elementary topos is local if the terminal object is
connected: implies or , and
projective: every epi admits a section .
Some authors have instead used the term “local” to refer just to the condition that is connected; note this is equivalent to being a local rig. In his thesis, Steve Awodey upgraded this, saying that a topos is hyperlocal if both connectivity and projectivity are satisfied. As argued above, for a Grothendieck topos this is slightly weaker than locality, although the difference is only in the inclusion of the trivial topos (for if then ). Note that a local elementary topos, as defined above, is constructively well-pointed if and only if is additionally a generator.
The free topos is a local elementary topos.
A geometric morphism is local precisely if
This is (Johnstone, theorem 3.6.1 vi)).
This appears in (Shulman).
This appears as (Johnstone, lemma C3.6.4).
Since in a local topos the global section functor is a left adjoint, it preserves epimorphisms. Since it is a right adjoint it preserves the terminal object. Therefore is an epimorphism in Set, hence a surjection, meaning that is inhabited. Since (see global section geometric morphism), the claim follows.
In a topos every epimorphism is an effective epimorphism. Therefore being an epi means that is a (-1)-connected object. Therefore the above statement says in terms of (infinity,1)-category theory that a non-trivial local topos has homotopy dimension 0.
The same is true for any local (infinity,1)-topos.
For any local topos , the base topos is equivalent to the category of sheaves for a Lawvere-Tierney topology on . A sound and complete elementary axiomatization of local maps of (bounded) toposes can be given in terms of properties of topos and topology (AwodeyBirkedal)
We discuss first
This appears under the term “principal” in (Awodey-Birkedal, def. 2.1).
We use the notation “” and “” oppositely to the use on p.14 of Awodey-Birkedal. Our convention is such that it harmonizes with the terminology at cohesive topos and cohesive (infinity,1)-topos, where it makes interpretational sense to pronounce “” as “flat”.
The left adjoints for all extend to a functor on all of .
This appears as (AwodeyBirkedal, lemma 2.3).
By the discussion at category of sheaves we have that is given by the composite
If now the left adjoint exists, it follows that this comes from a left adjoint to as
Therefore the -counit provides morphisms
whose image factorization we claim provides the least dense subobjects.
To show that is dense it is sufficient to show that
is an isomorphism.
is indeed an isomorphism.
Moreover, by one of the equivalent characterizations of reflective subcategories we have (…)
An object is called discrete if for all -local isomorphisms the induced morphism
is an isomorphism (of sets, hence a bijection).
An object is called o-discrete if .
Every discrete object is o-discrete.
Axiom 1. is essential.
Axiom 2 b. With as above, there is a discrete object and an epimorphism .
These axioms characterize local geometric morphisms .
If is fixed to be the terminal object (in which case Axiom 2 b becomes empty), then they characterize local and localic geometric morphisms.
This is (Awodey-Birkedal, theorem 3.1) together with the discussion around remark 3.7.
Notice that is the presheaf topos over the point category, the category with a single object and a single morphism. Therefore the constant presheaf functor
can be thought of as sending a set , hence a functor to the composite functor
where the vertical morphisms are , the point being that they exist for every given the presence of the terminal object. It follows that such a natural transformation is given by any and one and the same function .
Since adjoints are essentially unique, it follows that the global section functor is given by taking the limit, .
Observe that the terminal object is the initial object in the opposite category . But the limit over a diagram with initial object is given simply by evaluation at that object, and so we have for any that
hence that the global section functor is simply given by evaluating a presheaf on the terminal object of .
Limits and colimit in a presheaf category are computed objectwise over (see at limits and colimits by example). Therefore evaluation at any object in preserves in limits and colimits, and in particular evaluation at the terminal object does. Therefore preserves all colimits. Hence by the adjoint functor theorem it has a further right adjoint .
We can compute it explicitly by the Yoneda lemma and using the defining Hom-isomorphism of adjoints to be the functor such that for the presheaf is given over by
If is a topological space, or more generally a locale, then is local (over Set) iff has a focal point , i.e. a point whose only neighborhood is the whole space. In this case, the extra right adjoint to the global sections functor is given by the functor which computes the stalk at . This can also be given without reference to , by the formula
for sets and open subsets .
Let be a partial combinatory algebra and let be a sub partial combinatory algebra of . Then there is a (localic) local geometric morphism from the relative realizability topos? to the standard realizability topos .
Let denote the 2-category of local Grothendieck toposes (over Set) with all geometric morphisms between them. Let denote the 2-category whose objects are pointed toposes? (i.e. (Grothendieck) toposes equipped with a geometric morphism ), and whose morphisms are pairs such that is a geometric morphism and is a (not necessarily invertible) geometric transformation.
Note that if is a local topos with global sections geometric morphism , then the adjunction is also a geometric morphism . In this way we have a functor , which is a full embedding, and turns out to have a right adjoint: this right adjoint is called the localization of a pointed topos at its specified point. For example:
If is a small category and is an object of , then the localization of the presheaf topos at the point induced by can be identified with the presheaf topos over the over category of over . By the general properties of over toposes, this is equivalently the over-topos (where is regarded in by the Yoneda embedding).
If is the Zariski spectrum of a commutative ring , and is a prime ideal of (i.e. a point of ), then the localization of at can be identified with , where denotes the localization of at . Of course, this is the origin of the terminology.
A similar construction is possible for bounded toposes over any base (not just Set).
If is a tiny object then by definition we have
Inserting all this into the above pullback gives the pullback
Let be a commutative ring (such as or a field), let be the prime spectrum of , and let be the big Zariski topos for (i.e. the classifying topos for local $A$-algebras). For each element of , we have an open subset , and these open subsets constitute a basis for the topology on . The full subcategory of the frame of open subsets of spanned by these basic open subsets admits a contravariant full embedding in the category of finitely-presented -algebras via the functor (the well-definedness of this functor requires a non-trivial check!), and this functor moreover has the cover lifting property, so induces a local geometric morphism .
local topos / local (∞,1)-topos
Standard references include
and Chapter C3.6 of
A completely internal characterization of local toposes is discussed in
This is based on part 2 of
Free local constructions are considered in
Notions of local topos, with a view to logical completeness theorems, appear in Steve Awodey’s thesis: