The classifying topos of a geometric theory (that is, of a first-order theory whose axioms can be presented in the form , where and are first-order formulas obtained from atomic formulae by only using finitary conjunctions, infinitary disjunctions and existential quantifications and is a set of variables containing all the free variables of and ) is a Grothendieck topos which classifies the models of in Grothendieck toposes, in the sense that for any Grothendieck topos there is an equivalence of categories
natural in , where is the category of geometric morphisms from to and is the category of models of the theory in the topos . In other words, there is a model of in , called 'the' universal model of , characterized by the universal property that any model in a Grothendieck topos can be obtained, up to isomorphism, as a pullback of the model along the inverse image of a unique (up to isomorphism) geometric morphism from to , a sort of 'shadow' of . In the following picture, the big shapes represent different toposes while the inner lighter shapes represent models of a given theory inside them; in particular, the dark yellow star represent the classifying topos of a given theory and the light yellow diamond represent the universal model of the theory inside it.
It is a fact that most of the (first-order) theories naturally arising in Mathematics have a geometric axiomatization (over their signature). Anyway, if a finitary first-order theory is not geometric, we can canonically construct a coherent theory over a larger signature, called the Morleyization of , whose models in the category Set of sets (and, more generally, in any Boolean coherent category) can be identified with those of . The notion of Morleyization is important because it enables us to study any kind of first-order theory by using the methods of Topos Theory. In fact, we can expect many important properties of first-order theories to be naturally expressible as properties of their Morleyizations, and these latter properties to be in turn expressible in terms of invariant properties of their classifying toposes. For example, a first-order theory is complete if and only if its Morleyization satisfies the following property: any coherent (equivalently, geometric) sentence over its signature is provably equivalent to or to , but not both; and this property is precisely equivalent to saying that its classifying topos is two-valued.
It is often the case that one can 'turn' a finitary first-order theory into a geometric one by adding some sorts to the signature of the original theory and axioms in the extended language so to ensure that each of the first-order formulae which appear in the axioms of the original theory becomes equivalent to geometric formulae in the extended signature. Notice that the Morleyization represents a canonical way for doing this which works for any kind of first-order theory, but that one can nonetheless, in particular cases, obtain a geometric theory starting from a finitary first-order one whose set-theoretic models can be identified with those of the original theory in alternative ways, as indicated above.
The classifying topos of a geometric theory can be canonically built as the category of sheaves on the geometric syntactic category of the theory with respect to the geometric topology on it. For smaller fragments of geometric logic, such as for example Horn (resp. regular, coherent) logic, there exist variations of this syntactic construction, consisting in replacing the geometric syntatic site of the theory with its cartesian (resp. regular, coherent) syntactic site. In particular, any finitary algebraic theory can be realized as the topos of presheaves on its cartesian syntactic category.
There are also methods of different nature for 'calculating' the classifying topos of a geometric theory. For example, as we shall see below, every representation of a geometric theory as an extension of a given geometric theory over its language leads to a representation of its classifying topos as a subtopos of the classifying topos of the latter theory. In fact, more generally, it is reasonable to expect 'different ways of looking at a certain theory' to materialize into different representations of its classifying topos (see the discussion below).
We said above that to any geometric theory one can associate a site, namely the geometric syntatic site of the theory, such that its classifying topos is equivalent to the category of sheaves on this site. Conversely, every Grothendieck topos can be regarded, although not canonically, as the classifying topos of a geometric theory; indeed, if a topos is represented as the category of sheaves on a site then can be regarded as the classifying topos of the geometric theory of -continuous flat functors on .
The existence of Morita-equivalences between different theories can thus be interpreted geometrically as the existence of different sites of definition for a given topos. The representation theory for Grothendieck toposes in terms of sites therefore becomes an essential tool for investigating Morita-equivalences.
The unifying methodologies are based on the construction of topos-theoretic 'bridges' connecting distinct mathematical theories with each other.
If and are two Morita-equivalent theories (that is, geometric theories classified by the same topos) then their common classifying topos can be used as a 'bridge' for transferring information between them:
The transfer of information takes place by expressing topos-theoretic invariants (that is, properties or constructions on toposes which are stable under categorical equivalences) defined on their common classifying topos directly in terms of the theories and . This is done by associating to each of the two theories a site of definition for its classifying topos (for example, its geometric syntactic site) and then using site characterizations (that is, characterizations of topos-theoretic invariants on a topos in terms of its sites of definition, i.e. of the sites such that is categorically equivalent to ) to establish relationships between the resulting sites.
Specifically, suppose that and are two sites of definition for the same topos, and that is a topos-theoretic invariant. Then one can seek site characterizations for the invariant , that is (in the case is a property - the case of being a 'construction' admits an analogous treatment which we do not address in full generality here) a logical equivalence of the kind 'the topos satisfies if and only if satisfies a property (written in the language of the site )' and, similarly for , a logical equivalence of the kind 'the topos satisfies if and only if satisfies a property '.
Clearly, such characterizations immediately lead to a logical equivalence between the properties and of the two sites, which can then be seen as different manifestations of a unique property, namely , in the context of the two different sites and .
The crucial aspect here is that the 'bridge technique' allows to interpret and study a large part of dualities and equivalences arising in Mathematics through the investigation of the representation theory of Grothendieck toposes and the site-level characterization of topos-theoretic invariants. The theory of Grothendieck toposes and their representations thus becomes a sort of 'meta-theory of mathematical duality', which makes it possible to effectively compare distinct mathematical theories with each other and transfer knowledge between them.
In the following section we discuss more in detail the subject of Morita-equivalences, which play in our context the role of 'decks' of our 'bridges', and of site characterizations for topos-theoretic invariants, which constitute the 'arches' of our 'bridges'.
Two geometric theories and are Morita-equivalent if and only if they have equivalent categories of models in every Grothendieck topos , naturally in , that is if for each Grothendieck topos there is an equivalence of categories
such that for any geometric morphism the following diagram commutes (up to isomorphism):
Of course, the classifying topos being the representing object for the semantics (pseudo)functor of the theory , two theories are Morita-equivalent if and only if they have the same classifying topos (up to equivalence).
Given the level of technical sophistication of this definition, it is reasonable to wonder if Morita-equivalences naturally arise in Mathematics and, in case, if there are systematic ways for 'generating' them. The following remarks are meant to show that the answer to both these questions is positive.
If two geometric theories and have equivalent categories of models in the category Set then, provided that the given categorical equivalence
is established by using only constructive logic (that is, by avoiding for instance the law of excluded middle and the axiom of choice) and geometric constructions (that is, by only using set-theoretic constructions which involve finite limits and small colimits, equivalently which admit a syntactic formulation involving only equalities, finite conjunctions, (possibly) infinitary disjunctions and existential quantifications), it is reasonable to expect the original equivalence to 'lift' to a Morita-equivalence
between and .
[Indeed, a Grothendieck topos behaves logically as a 'generalized
universe of sets' in which one can perform most of the classical
set-theoretic constructions with the only significant exception of
arguments requiring non-constructive principles, and hence we can
naturally expect to be able to generalize the original equivalence
between the categories of set-based models of the two theories to
the case of models in arbitrary Grothendieck topos; and the fact
that the constructions involved in the definition of the equivalence
are geometric ensures that the naturality condition for
Morita-equivalences mentioned above is satisfied (since geometric
constructions are preserved by inverse image functors of geometric
morphisms, so that one can
globally obtains a Morita-equivalence.
See Exercises VIII.7-8-9 in
this book for an example of application of this general method
to the case of the equivalence between the category of models of the
theory of decidable (resp. infinite decidable) objects and the
theory of flat (resp. flat and continuous with respect to the atomic
topology) functors on the opposite of the category of finite sets
and injections between them. A more general investigation of of this
method in the context of theories of presheaf type is pursued in
this paper, leading to a criterion for a geometric theory to be
of presheaf type which is applicable to a wide class of theories
whose finitely presentable models are finite.]
Two cartesian (in particular, finitary
algebraic) theories
and
have equivalent categories of models in
Set if and only if they are Morita-equivalent (or,
equivalently, if and only if their cartesian syntactic categories
are equivalent).
[Indeed,
if and only if the cartesian syntactic categories of
and
are
equivalent (since for any such theory its category of models in
Set is equivalent to the category of cartesian functors
from its syntactic category to Set), if and only if the classifying toposes of
and
(which are equivalent respectively to the presheaf topos on the
cartesian syntactic category of
and to
the presheaf topos on the cartesian syntactic category of
) are
equivalent.]
If two geometric (resp. coherent) theories
have equivalent geometric (resp. coherent) syntactic categories then
they are Morita-equivalent.
[Indeed, the classifying topos can be constructed as the category of
sheaves on the syntactic site of the theory and the Grothendieck
topologies on the syntactic sites can be defined intrinsically in
terms of the categorical structures on the underlying categories.]
If two finitary first-order theories are
biinterpretable (in the sense of classical Model Theory) then their
Morleyizations are Morita-equivalent.
[Indeed, the coherent syntactic category
of the Morleyization of a first-order theory is equivalent to the
first-order syntatic category of the theory.]
Two associative rings with unit are
Morita-equivalent (in the classical, ring-theoretic, sense) if and
only if the algebraic theories axiomatizing the (left) modules over
them are Morita-equivalent (in the topos-theoretic sense). In fact,
two rings are Morita-equivalent if and only if the cartesian
syntactic categories of these theories are equivalent.
[Indeed, by the first remark above, these
theories are Morita-equivalent if and only if their categories of
set-based models are equivalent (equivalent, if and only if their
cartesian syntactic categories are equivalent), that is if and only if the
categories of (left) modules over the two rings are equivalent.
Specifically, for each ring R the theory axiomatizing its (left)
R-modules can be defined as the theory obtained from the algebraic
theory of abelian groups by adding one unary function symbol for
each element of the ring and writing down the obvious axioms
involving the terms formed with such symbols as well as with the
symbol for addition in the group which express the conditions in the
definition of R-module.]
Other notions of Morita-equivalence for
various kinds of algebraic or geometric structures considered in the
literature can be reformulated as equivalences between different
representations of the same topos, and hence as Morita-equivalences
between different geometric theories. For instance:
- Two topological groups are
Morita-equivalent (in the sense of
this paper) if and only if the toposes of continuous actions
over them are equivalent. (A natural analogue of this notion for
topological and localic groupoids has been studied by several
authors and a summary of the main results in contained in section
C5.3 of
this
book).
- Two small categories are
Morita-equivalent (in the sense of
this paper) if and only if the corresponding presheaf toposes
are equivalent, that is if and only if their Karoubian completions
are equivalent.
- Two inverse semigroups are
Morita-equivalent (in the sense of
this paper or, equivalently, of
this paper) if and only
if their classifying toposes (as defined in
this
paper) are equivalent (cf.
this paper).
Categorical dualities or
equivalences
between 'concrete' categories can often be seen as arising from the
process of 'functorializing' Morita-equivalences which express
structural relationships between each pair of objects corresponding
to each other under the given duality or equivalence (cf. for
example this paper and
this paper).
[In fact, the theory of geometric
morphisms of toposes provides various natural ways of
'functorializing' bunches of Morita-equivalences.]
As a matter of fact, the notion of Morita-equivalences
materializes in many situations the intuitive feeling of 'looking at
the same thing in different ways', meaning, for instance, describing the same
structure(s) in different languages or constructing a given object
in different ways.
[Concrete examples of this general remark
can be found for instance in
this paper, where the different
constructions of the Zariski spectrum of a ring, of the
Gelfand spectrum of a C*-algebra, and of the Stone-Cech compactification of a topological
space are interpreted as Morita-equivalences between different
theories.]
Different ways of looking at a given mathematical theory can often
be formalized as Morita-equivalences.
Indeed, different ways of describing the
structures axiomatized by a given theory can often give rise to a
theory written in a different language whose models (in any
Grothendieck topos) can be identified, in a natural way, with those
of the former theory and which is therefore Morita-equivalent to it.
A geometric theory alone generates an
infinite number
of Morita-equivalences, via its 'internal dynamics'. In fact, any
way of looking at a geometric theory as an exension of a geometric
theory written in its signature provides a different representation
of its classifying topos, as a subtopos of the classifying topos of
the latter theory (cf. the duality
theorem).
[An intuitive image which may help
understand this remark is the following. We can think of a
mathematical theory as a living organism which is born when its
original axioms are given and which develops in time through the
discovery of new results in it. Any collection of results which are
already known to hold in the theory can be taken as axioms of a new
theory of which the former theory can be viewed as an extension and
hence leads, by the remark above, to a new representation of its
classifying topos. Any splitting between the already known (say, the
'past' of the theory) and the potentially true (that is, the
'future' of the theory) thus gives rise to a Morita-equivalence of
the theory with another one (via the duality theorem).]
Different sites of definition for a given topos can be interpreted logically as Morita-equivalences between different theories (in fact, the converse also holds, in the sense that any Morita-equivalence gives canonically rise to two different sites of definition of the common classifying topos). The representation theory of Grothendieck toposes in terms of sites (and, more generally, any technique that one may employ for obtaining a different site of definition for a given topos) thus constitutes an effective tool for generating Morita-equivalences. For example, (a version of) Grothendieck's Comparison Lemma asserts that for any site and any full subcategory of which is -dense (that is, for every object of there exists a -covering sieve on it generated by arrows whose domains lie in ), we have an equivalence , where is the Grothendieck topology on induced by .
Different separating sets for a given topos (recall that a separating set for a topos is a set of objects of the topos such that any object in the topos can be covered by a family of arrows whose domains belong to the set) give rise to different sites of definition for it; indeed, for any separating set of objects of a Grothendieck topos , we have an equivalence , where is the Grothendieck topology on induced by the canonical topology on . Notice incidentally that all the sites of the form are subcanonical, and any subcanonical site of definition for the topos is, up to canonical equivalence, of this form. In particular, for any topological space and any basis for it, we have an equivalence (as we shall see in this section and this other one, the most interesting Morita-equivalences arise when the topology can be characterized intrinsically in terms of ).
As we remarked above, the bridge technique is based on the existence of site characterizations for topos-theoretic invariants, that is criteria connecting invariant properties (resp. constructions) on toposes and properties (resp. constructions) of their sites of definition (written in their respective languages).
It thus becomes crucial to investigate the behaviour of topos-theoretic invariants with respect to sites. As a matter of fact, such a behaviour is often very natural, in the sense that topos-theoretic invariants of whatever kind generally admit natural site characterizations, which explains the technical effectiveness of the 'bridge method'.
For instance, bijective characterizations for a wide class of geometric invariants of toposes, notably including the property of a topos to be atomic (resp. locally connected, localic, equivalent to a presheaf topos, compact, two-valued) were obtained in this paper.
Also, several notable invariants of subtoposes were shown in this paper to admit natural site chacterizations as well as explicit logical descriptions in terms of the theories classified by the relevant toposes.
Moreover, it was shown in this paper that a wide class of logically-inspired invariants of topos, obtained by interpreting first-order formulae written in the language of Heyting algebras, admit natural and bijective site characterizations.
Topos-theoretic invariants relevant in Algebraic Geometry and Homotopy Theory, such as for example the cohomology and homotopy groups of toposes, are also well-known to admit, at least in many important cases, natural characterizations in terms of sites.
The startling aspect in all of this is that, while it is often possible to obtain, by using topos-theoretic methods, site characterizations for topos-theoretic invariants holding for wide classes of sites, such criteria can be highly non-trivial as far as their mathematical depth is concerned (since the representation theory of toposes is by all means a non-trivial subject) and hence, when applied to specific Morita-equivalences, can lead to deep results on the relevant theories (especially when the given Morita-equivalence is a non-trivial one).
In fact, these insights can be very surprising, when observed from a concrete point of view (that is, from the point of view of the two sites related by the Morita-equivalence), since a given topos-theoretic invariant may manifest itself in very different ways in the context of different sites. For example, for any site the topos satisfies the invariant property to be De Morgan if and only if for any object of the category and any -closed sieve on the sieve
is -covering, where (for any ).
In particular, such property specializes, on a presheaf topos, to the condition on the underlying category to satisfy the dual of the amalgamation property (that is, the property that any two pairs of arrows with common codomain can be completed to a commutative square), while on a topos of sheaves on a topological space it specializes to the property of the space to be estremally disconnected. Further examples of the behavior of topos-theoretic invariants with respect to sites are provided in this section.
In light of these results, it is reasonable to expect that many more topos-theoretic invariants that one might introduce will admit natural site characterizations; in fact, one of the purposes of the unification programme is to document the behaviour of topos-theoretic invariants in relation to sites and develop general methods for obtaining site characterizations for new as well as known invariants.
Notice that the unifying method based on the view 'toposes as bridges' generates results transverally to the various mathematical fields, in a uniform and effective way which is determined by the form of the toposes involved and the invariants considered on them.
It is worth to remark that, unlike the traditional, 'dictionary-oriented' method of translation based on a 'renaming', according to the given 'dictionary', of the primitive constituents of the information across the two different languages, our 'invariant-oriented' method of translation of results across different mathematical theories based on topos-theoretic bridges proceeds through a 'structural unraveling' of appropriate invariants across different representations of the toposes involved, rather than through the use of an explicit description of the Morita-equivalence serving as 'dictionary'. In fact, for the transfer of 'global' properties of toposes, it is only the existence of a Morita-equivalence that really matters, and we can well ignore its actual definition (since, by definition, a topos-theoretic invariant is stable under any kind of categorical equivalence); of course, if one wants to establish more 'specific' results, an explicit description of the Morita-equivalence becomes necessary (in which case, one can use invariant properties of objects of toposes rather than invariants properties of the 'whole topos') but for treating most of the 'global' properties of theories this is not at all necessary.
We should also remark that the arches of a given bridges need not necessarily be 'symmetric', that is arising from the instantiation in the context of two given sites of a unique site characterization holding for both of them. Indeed, for any given Morita-equivalence and topos-theoretic invariant, the invariant can admit a certain site characterizations holding for sites of a given form including one of the sites involved in the Morita-equivalence but not the other, which constitute one of the arches of the bridge, and another site characterization holding in particular for the other site, yielding the other arc of the bridge. As an example, take the property of a topos to be coherent: this property does not admit an 'elementary' bijective site characterization holding for all sites, but it admits a bijective site characterization holding for all trivial sites (i.e. sites in which the Grothendieck topology is trivial) and an implicative characterization of the kind 'if a theory is coherent, then its classifying topos is coherent', which can be combined together to obtain a bridge expressing a theorem about theories of presheaf type.
As an illustration of the natural behaviour of topos-theoretic invariants with respect to sites and geometric theories, we focus in the next section on the notion of subtopos.
The notion of subtopos is a topos-theoretic invariant which has a natural behaviour with respect to sites. Indeed, by a fundamental result in Topos Theory, for any site the subtoposes of a Grothendieck topos are in bijective correspondence with the Grothendieck topologies on which contain . Considered on the classifying topos of a geometric theory, this characterization leads to a description of these subtoposes as classifying toposes for theories which extend the given theory over its signature.
Specifically, given a geometric theory over a signature , we say that a geometric theory over is a quotient of if every geometric sequent over which is provable in is provable in (we shall consider quotients up to the equivalence relation which identifies two quotients when they prove exactly the same geometric sequents over their signature). Then we have the following duality theorem (Theorem 3.6 in this paper).
Theorem: For any geometric theory , the subtoposes of its classifying topos are in natural bijection with the quotients of . Moreover, under this assignment, any quotient of corresponds to its classifying topos.
Notice that this result can be interpreted as a site characterization for the property of being a subtopos on a given classifying topos in terms of the geometric syntactic site of a theory classified by it.
As shown in this paper, the theorem allows to transfer a lot of information known in Topos Theory about subtoposes into Logic; for instance, the collection of all subtoposes of a given topos is known to have the structure of a coHeyting algebra, (with the natural ordering on them), a structure which can be transferred at the level of sites and of theories (in the latter case across the bijection provided by the theorem). In fact, the lattice operations on Grothendieck topologies admit explicit and natural descriptions which can be used, among the other things, to obtain a logical characterizations of the Heyting algebra structure present on the collection of all the quotients of a given geometric theory.
The following picture is meant to represent the lattice structure on the collection of subtoposes of a given topos and the isomorphic structures (via the duality theorem) on the collection of quotients of different theories (, , , ...) classified by .
A truly remarkable fact is that the abstract topos-theoretic notions defined at the level of subtoposes, once transferred to the level of theories via the duality theorem, are often of very natural logical or mathematical interest.
For example, the subtopos arising as the surjection-inclusion factorization of the geometric morphism to the classifying topos of a theory corresponding to a model of in can be characterized as the classifying topos of the quotient of consisting of all the geometric sequents over the signature of which hold in the model .
Another example is given by the notion of DeMorganization (resp. Booleanization) of a subtopos (as defined in this paper), whose logical interpretation defines a natural operation on theories. For instance, as shown in this paper, the DeMorganization of the (coherent) theory of fields can be identified with the theory of fields of finite characteristic which are algebraic over their prime fields, while its Booleanization can be identified with the theory of fields of finite characteristic which are algebraic over their prime fields as well as algebraically closed.
Let us now discuss the duality theorem in the context of the 'bridge-building' technique.
If we take two Morita-equivalent theories and associate to them their geometric syntactic sites then the syntactic characterizations given by the duality theorem yield a 'bridge' providing a bijection between the quotients of the two theories (written over their respective signatures). This result is interesting because, when specialized to concrete theories, it can be rather deep, and because it represents an instance of the unification between semantics (which is the context in which Morita-equivalences arise) and syntax (which is the context in which quotients lie) given by the theory of classifying toposes. Indeed, while in the setting of classical model theory one does not have, for general first-order theories, models of syntactic nature which enjoy a strong form of completeness, with the consequence that the semantics remain separate from the syntax and hence that it is not guaranteed that semantic equivalences of theories restrict to semantic equivalences for quotients of them (so that problems of the kind mentioned above are generally rather hard to approach), in the topos-theoretic context the universal models lying in classifying toposes satisfy a strong form of completeness, by virtue of the fact that they admit a purely syntactic description.
Another interesting context for the application of the duality theorem in connection to the philosophy 'toposes as bridges' is given by the class of theories of presheaf type. A geometric theory is said to be of presheaf type if it is classified by a presheaf topos. One can prove (see this paper) that a theory is of presheaf type if and only if it is classified by the topos
where is (a skeleton of) the category of finitely presentable models (that is, the models such that their covariant hom-functor preserves filtered colimits).
For any theory of presheaf type we thus have a Morita-equivalence
where is the geometric syntactic site of the theory .
Considering the notion of subtopos as an invariant, and recalling the site characterizations for this invariant obtained above (and in particular the duality theorem), we obtain a 'bridge' between the two different sites of definition yielding a bijection between the quotients of the theory and the Grothendieck topologies on the opposite of the category . For each quotient of we thus have a (unique) Grothendieck topology on such that the given equivalence restricts (along the obvious geometric inclusions) to an equivalence
Notice that the topology depends not only on the quotient but also on the original equivalence of toposes
since the composite of a geometric inclusion to a certain topos with an equivalence on that topos is in general non-isomorphic to the original inclusion. Nonetheless, for each theory of presheaf type , one can always put an equivalence
in a canonical form, and hence explicitly describe the relationship between the quotients and their associated topologies ; specifically, one can suppose, without loss of generality, that for any theory of presheaf type , the equivalence
is induced by the universal property of as a classifying applied to the model of in given by the forgetful functor from to Set (cf. this paper for more details). We shall say that the given Morita-equivalence for is canonical if it is of this form.
Any topos-theoretic invariant considered on a Morita-equivalence
thus gives rise, together with its site characterizations, to a 'bridge'
between the two different representations of the classifying topos.
An example of the insights which can be obtained by applying this method, we have the following theorem, which exhibits relations between 'geometric' properties of the site and logical properties of the theory . Of course, each topos-theoretic invariant gives rise to a different relation, and one can establish new relations by applying the same general method to other topos-theoretic invariants. The invariants which give rise to the following theorem are the property of a topos to be equivalent to a presheaf topos (resp. to be atomic, to be locally connected) and the property of an object of a topos to be irreducible (resp. an atom, indecomposable), and the site characterizations for them which are employed to derive the results are provided by this paper.
Theorem. Let be a quotient of a theory of presheaf type over a signature corresponding to a Grothendieck topology on . Then
If
is trivial then
satisfies the property that there exists a collection
of
-irreducible
geometric formulae-in-context over such that
for any geometric formula
over
, there
exist formulae in
(for ) and
-provably
functional geometric formulae
from the to
such that
is provable in
(where by -irreducible
formula we mean a geometric formula
such that
for any family
of
-provably
functional geometric formulae from
to
with the
property that
is provable in
,
there exist
and a -provably
functional geometric formula
from
to
such that
is provable in
).
Moreover, if the Morita-equivalence for
is canonical then every formula which presents a
-model is
-irreducible.
If the category
satisfies
the amalgamation property and
is the atomic
topology on the opposite of it then
satisfies the property that every geometric formula over its
signature is
-provably
equivalent to a disjunction of
-complete
formulae in the same context (where by a
-complete
formula we mean a geometric formula
such that
the sequent
is
not provable in
and
for every geometric formula
in the same
context either
or
is
provable in
).
Moreover, if the Morita-equivalence for
is canonical then every formula which presents a
-model is
-complete.
If the site
is
locally connected (that is, every
-covering sieve
is connected as a subcategory of the corresponding slice category)
then
satisfies the property that if for any geometric formula
over
there
exists a (unique) family
of
--indecomposable
geometric formulae in the same context such that
for each ,
-provably
implies ,,
for any distinct
,
is provable in
,, and
is provable in
(where by a
-indecomposable
geometric formula we mean a formula
such that
for any family
of
geometric formulae in the same context such that for each
,
-provably
implies and
for any distinct
,
is provable in
, we
have that
provable in
implies
p
provable in
for
some ).
Moreover, if the Morita-equivalence for
is canonical then every formula which presents a
-model is
-indecomposable.
This theorem might seem at first sight rather abstract and remote from the usual 'mathematical practice', but in fact concrete (and non-trivial) results about specific theories can be easily derived from it. For example, the theory of linear orders, as well as the theory of fields of finite characteristic which are algebraic over their prime fields, satisfy the first condition in the statement of the theorem, while the theory of atomless Boolean algebras, of infinite decidable objects, and of dense linear orders without endpoints satisfy the second condition. Please see this paper and this paper for more details.
Further concrete applications of the 'bridge method' are given in this section.
Finally, we show that it is possible to establish, under the hypothesis that the Morita-equivalence
for the theory be canonical, a precise description of the relation between the quotients of and the Grothendieck topologies on corresponding to them via the method above. Specifically, one can characterize the models of in Grothendieck toposes among the models of directly in terms of the Grothendieck topology through a categorical, semantic, condition, and describe an axiomatization of the theory directly in terms of and .
As shown in this paper, in any Grothendieck topos the -models coincide with the -models which are -homogeneous (in the sense of that paper). In particular, the models of in Set are precisely the models of in Set such that for any finitely presentable -model , any sieve and any arrow in there exist an arrow in and an arrow in such that :
In particular, if the category satisfies the amalgamation property (that is, the property that any pair of arrows with common domain can be completed to a commutative square) then one can consider the atomic topology on it (that is, the Grothendieck topology on it such that its covering sieves are precisely the non-empty ones), and the notion of -homogeneous model specializes to that of homogeneous model, that is of model of in Set such that for any arrow in and any arrow in there exists an arrow in such that . An illuminating application of the 'bridge technique' in connection to the notion of homogeneous model is described in this section.
To obtain an explicit axiomatization of the quotient in terms of , we exploit the fact, established in this section, that the finitely presentable models of correspond to the -irreducible formulae. From this it follows, in light of the duality theorem, that is equal to the quotient of axiomatized by the sequents of the form , where and are -irreducible formulae and is a family of -provably functional geometric formulae from to with the property that, regarded as a sieve on in , it is sent by the equivalence to a -covering sieve on .
The above-mentioned explicit construction of classifying toposes of quotients of theories of presheaf type via the the duality theorem can be regarded as a 'semantic' approach to the contruction of classifying toposes (as opposed to the syntactic approach consisting on taking the topos of sheaves on an appropriate syntactic site for the theory), which is particularly amenable for the applications in 'concrete' mathematical domains other than logic.
For example, by using this method one can calculate the classifying topos for the theory of local rings (namely, the Zariski topos) and the classifying topos for the theory of integral domains, both regarded as quotient of the theory of commutative rings with unit (which is of presheaf type being algebraic). Specifically, the classifying topos for the theory of commutative rings with unit, axiomatized in the natural way over the one-sorted the signature consisting of two binary function symbols and , one unary function symbol and two constants and , is the presheaf topos , where is the category of finitely generated (equivalently, finitely presented) rings and ring homomorphisms between them. The theory of local rings can be axiomatized as the quotient of this theory obtained by adding the axioms
and
,
while the theory of integral domains is the quotient obtained by adding the axioms
and
.
The classifying topos for the theory of local rings can be represented as the topos , where is the Grothendieck topology on defined as follows: given a cosieve in on an object , if and only if contains a finite family of canonical inclusions in where is any set of elements of which is not contained in any proper ideal of .
On the other hand, the classifying topos for the theory of integral domains can be represented as the topos , where is the Grothendieck topology on defined as follows: given a cosieve in on an object , if and only if either is the zero ring and is the empty sieve on it contains a non-empty finite family of canonical projections in where is any set of elements of such that .
Notice how the form of the two Grothendiek topologies and on the category reflects the form of the axioms of the two theories.