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Discrete and concrete objects
Local geometric morphism and relative local topos
A local geometric morphism between toposes is
a geometric morphism
such that a further right adjoint exists
and such that one, hence all, of the following equivalent conditions hold:
- The right adjoint is an -indexed functor.
- is connected, i.e. is fully faithful.
- The right adjoint is fully faithful.
- The right adjoint is cartesian closed.
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 .
This is especially common when Set, in which case the final condition is automatic since all functors are -indexed. Hence in that case we have the following simpler definition.
A sheaf topos is a local topos if the global section geometric morphism has a further right adjoint , making an adjoint triple
(As just stated, it is automatic in the case over that this is furthermore a full and faithful functor.)
A geometric morphism is local precisely if
there exists a geometric morphism such that ;
for every other geometric morphism the composite is an initial object in the hom-category of the slice 2-category of Topos over .
This is (Johnstone, theorem 3.6.1 vi)).
This appears in (Shulman).
This appears as (Johnstone, lemma C3.6.4).
Every local topos comes with a notion of concrete sheaves, a reflective subcategory which factors the topos inclusion of :
and is a quasitopos. See concrete sheaf for details.
Since a local geometric morphism has a left adjoint in the 2-category Topos, it is necessarily a homotopy equivalence of toposes.
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
Let be an elementary topos equipped with a Lawvere-Tierney topology .
Write for the -closure operation on subobjects , the sharp modality
for the reflective subcategory of j-sheaves.
We say that is an essential topology if for all objects the closure operation on posets of subobjects has a left adjoint :
This appears under the term “principal” in (Awodey-Birkedal, def. 2.1).
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
where the first morphism is sheafification and the second is full and faithful.
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.
Composing this morphism with of the -unit on (which is an isomorphism since is a full and faithful functor by the discussion at fully faithful adjoint triples) and using the triangle identity we have
Using that also is full and faithful and then 2-out-of-3 for isomorphisms it follows that hence
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.
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
Notice that in the presence of a terminal object in , is a full and faithful functor: a natural transformation has components
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 .
The functor has a left adjoint and a right adjoint, and these are – essentially by definition – the colimit and the limit operations
which send a presheaf/functor to its colimit or limit , respectively.
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
So in conclusion we have an adjoint triple where is a full and faithful functor. By the discussion at fully faithful adjoint triples it follows then that also is full and faithful.
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 .
Sheaves on a local site
For a local site, the category of sheaves is a local topos over .
For instance CartSp is a local site. Objects in are generalized smooth spaces such as diffeological spaces. The further right adjoint
is the functor that sends a set to the diffeological space on that set with codiscrete smooth structure (every map of sets is smooth).
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).
We check that the global section geometric morphism preserves colimits. It is given by the hom-functor out of the terminal object of , which is :
The hom-sets in the over category are fibers of the hom-sets in : we have a pullback diagram
Moreover, overserve that colimits in the over category are computed in .
If is a tiny object then by definition we have
Inserting all this into the above pullback gives the pullback
By universal colimits in the topos Set, this pullback of a colimit is the colimit of the separate pullbacks, so that
So does commute with colimits if is tiny. By the adjoint functor theorem then the right adjoint does exist and so is a local topos.
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 .
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
- Lars Birkedal, Developing Theories of Types and Computability via Realizability PhD Thesis (pdf)
Free local constructions are considered in