At the beginning of my Ph.D. studies, I had the intuition that Grothendieck toposes could serve as sorts of 'bridges' for effectively transferring information between distinct mathematical theories.
The results obtained in my Ph.D. thesis "The Duality between Grothendieck Toposes and Geometric Theories" actually provided compelling technical evidence for the validity of such a view, in the form of non-trivial applications pertaining to different fields of Mathematics, and stimulated new developments of the original idea, which eventually led to the general methodologies described in the programmatic paper "The unification of Mathematics via Topos Theory".
The principles introduced in this work are abstract and transversal to various mathematical fields, and the application of them can lead to surprising and non-trivial results in several areas of Mathematics; in fact, these methods have already generated ramifications into distinct branches including Algebra, Topology, Algebraic Geometry, Functional Analysis, Model Theory and Proof Theory, and the potential of this theory has just started to be explored.
My technical contributions, arising from an implementation of this philosophy in a variety of different mathematical contexts, can be described as follows; see also the synthesis text "Grothendieck toposes as unifying 'bridges' in Mathematics" which I have recently successfully defended as mémoire pour l'obtention de l'habilitation à diriger des recherches at the University of Paris 7.
My Ph.D thesis has addressed the long-standing problem of understanding the nature of the relatioship between Grothendieck toposes and geometric theories given by the theory of classifying toposes. The dissertation contains several new theorems on the relationship between geometric theories and their classifying toposes, in the form of criteria characterizing the geometric theories classified by relevant classes of toposes, but, most importantly, provides results and methods which justify the view of classifying toposes as 'bridges' mentioned above.
My Ph.D. work resulted in a series of eight papers, which contain theoretical results as well as 'concrete' applications of the philosophy 'toposes as bridges' in a variety of different mathematical contexts.
I provide a brief description of the contents of these
papers as
well as
of the later ones below.
In
the
paper
"Yoneda representations
of flat functors and classifying toposes"
I established a
fundamental
connection between topos theory and model theory leading to a number of
new developments and applications. The main result of this paper is an
explicit model-theoretic description of the theory classified by a given
topos of sheaves on a site in terms of the models of the theory
classified by the corresponding presheaf topos.
In the paper
"Fraïssé's
construction from a topos-theoretic perspective"
I produced a topos-theoretic interpretation of Fraïssé’s construction in
model theory, leading to new results in logic, in particular on
countably categorical theories. Specifically, the three concepts
involved in Fraisse’s constructions are seen to correspond precisely to
three different ways for looking at the same classifying topos according
to the philosophy
'toposes as bridges'.
In the paper
"De
Morgan classifying toposes"
I introduced a general method for deciding whether a Grothendieck topos
satisfies De Morgan’s law (resp. the law of excluded middle) or not,
with various applications to the theory of classifying toposes. In
particular, I provided complete characterizations of the classes of
theories classified by a De Morgan (resp. a Boolean) topos and
introduced the notions of Booleanization and DeMorganization of a
geometric theory. In this context I also exhibited equivalences of
well-known logical notions (those of De Morgan and Boolean algebra) with
topological ones.
In my paper with Peter Johnstone
"De
Morgan’s law and the theory of fields"
we provided applications in algebra of a new notion (that of
DeMorganization of a topos) introduced by myself in the paper
"De
Morgan classifying toposes";
specifically, we have shown that the DeMorganization of the classifying
topos for the theory of fields classifies the theory of fields of
nonzero characteristic which are algebraic over their prime fields.
In
the
paper
"Atomic toposes and
countable categoricity"
I established several results on atomic
toposes and gave
a model-theoretic characterization of the class of geometric theories
classified by an atomic topos having enough points; in particular, I
showed that every complete geometric theory classified by an atomic
topos is countably categorical.
The central result of the paper "Lattices of theories" is a duality theorem providing a central connection between geometric logic and the theory of Grothendieck toposes and paving the way for a transfer of ideas and results from the theories of elementary and Grothendieck toposes to logic. Specifically, the theorem provides, for any geometric theory, a bijection between the subtoposes of its classifying topos and the geometric extensions of the theory (considered up to a natural notion of syntactic equivalence). The paper gives two different proofs of the theorem and investigates the logical counterparts of many constructions and results in elementary topos theory which apply to subtoposes; the resulting notions are seen to be of natural logical and mathematical interest and several applications of the general theory are given. The results of this paper have also led to a solution to the third of the "Open problems in Topos Theory" by F. W. Lawvere.
The paper
"Universal
models and definability"
consists in a systematic investigation of universal models in Topos
Theory
with a particular emphasis on their applications to definability by
geometric formulae and to the investigation of the law of excluded
middle and De Morgan’s law on Grothendieck toposes.
The paper "Syntactic characterizations of properties of classifying toposes" provides characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (respectively connected and locally connected, atomic, compact, presheaf) topos, and exploited the existence of multiple sites of definition for a given topos to establish various results on quotients of theories of presheaf type.
In
the
paper
"A
characterization theorem for geometric logic",
I gave a semantic characterization of the classes of structures arising
as the classes of models of a geometric theory inside Grothendieck
toposes, solving in particular a twenty-year old problem of Ieke
Moerdijk.
The paper
"The
unification of Mathematics via Topos Theory"
is
a sort of
'scientific
manifesto'
of my research programme, in which I advocate that Grothendieck toposes
can serve as unifying spaces being able to act as
'bridges'
for effectively transferring knowledge between distinct mathematical
theories. A general overview
The
paper "A topos-theoretic
approach to Stone-type dualities"
provides
a general topos-theoretic perspective on Stone-type dualities.
The theory described in this paper arises from a faithful implementation
of the philosophy ‘toposes as bridges’, and
consists in
a general framework for analyzing the known Stone-type dualities, as
well as for introducing new ones: several known dualities are seen to be
specializations of just one topos-theoretic phenomenon, and new
dualities and equivalences between categories of preorders and
categories of posets,
locales or topological spaces are established. The paper also contains a
variety of other applications of the philosophy ‘toposes as bridges’ in
Algebra, Algebraic Geometry, Topology and Logic.
In the paper "A general method for building reflections" we introduce a general technique for generating reflections between categories. The method can be profitably applied to generate adjunctions starting from geometric morphisms between Grothendieck toposes; as particular cases, we recover various well-known Stone-type adjunctions and establish several new ones.
The paper "Priestley-type dualities for partially ordered structures" provides a general framework for generating dualities between categories of partial orders and categories of ordered Stone spaces; the classical Priestley duality for distributive lattices is recovered as a particular case and several other dualities for different kinds of partially ordered structures are established.
In the paper "Gelfand spectra and Wallman compactifications" we carry out a systematic, topos-theoretically inspired, investigation of Wallman compactifications with a particular emphasis on their relations with Gelfand spectra and Stone-Cech compactifications. In addition to proving several specific results about Wallman bases and maximal spectra of distributive lattices, we establish a general framework for functorializing the representation of a topological space as the maximal spectrum of a Wallman base for it, which allows to generate different dualities between categories of topological spaces and subcategories of the category of distributive lattices; in particular, this leads to a categorical equivalence between the category of commutative C*-algebras and a natural category of distributive lattices. We also establish a general theorem concerning the representation of the Stone-Cech compactification of a locale as a Wallman compactification, which subsumes all the previous results obtained on this problem.
In the paper "Topologies for intermediate logics" we investigate the problem of characterizing the classes of Grothendieck toposes whose internal logic satisfies a given assertion in the theory of Heyting algebras, and introduce natural analogues of the double negation and De Morgan topologies on an elementary topos for a wide class of intermediate logics.
In the paper “Topological Galois Theory” we introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of continuous actions of a topological group. Our framework subsumes in particular Grothendieck's Galois theory and allows to build Galois-type equivalences in new contexts, such as for example graph theory and finite group theory.
In the paper "The Morita-equivalence between MV-algebras and abelian l-groups with strong unit" (joint with A. C. Russo), we show that the theory of MV-algebras is Morita-equivalent to that of abelian l-groups with strong unit. This generalizes the well-known equivalence between the categories of set-based models of the two theories established by D. Mundici in 1986, and allows to transfer properties and results across them by using the methods of topos theory. We discuss several applications, including a sheaf-theoretic version of Mundici's equivalence and a bijective correspondence between the geometric theory extensions of the two theories.
In the paper "Extensions of flat functors and theories of presheaf type", we develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos.
In the paper "Cyclic theories" (joint with N. Wentzlaff) we describe a geometric theory classified by Connes-Consani's epicylic topos and two related theories respectively classified by the cyclic topos and by the topos of presheaves on the monoid of non-zero natural numbers.
In the paper "Lattice-ordered abelian groups and perfect MV-algebras" (joint with A. C. Russo) we establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-intepretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.
In the paper "General affine adjunctions, Nullstellensätze, and dualities" (joint with V. Marra and L. Spada) We develop an abstract categorical framework that generalises the classical "system-solution" adjunction in algebraic geometry, proving that such adjunctions take place in a multitude of contexts. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the theory of dualities generated by a "schizophrenic" object. Notably, classical dualities like Stone duality for Boolean algebras, Gelfand duality for commutative C*-algebras, Pontryagin duality for Abelian groups, turn out to be special instances of this framework. To determine how such general adjunctions restrict to dualities we prove abstract analogues of Hilbert's Nullstellensatz and Gelfand-Kolmogorov-Stone lemma, completely characterising the fixed points on one side of the adjunction.
In the paper "Syntactic
categories for Nori motives" (joint with L. Barbieri-Viale and L.
Lafforgue), we give a new construction, based on categorical logic, of
Nori's
In the paper "Motivic toposes" we present a research programme aimed at constructing classifying toposes of Weil-type cohomology theories and associated categories of motives, and introduce a number of notions and preliminary results already obtained in this direction. In order to analyze the properties of Weil-type cohomology theories and their relations, we propose a framework based on atomic two-valued toposes and homogeneous models. Lastly, we construct a syntactic triangulated category whose dual maps to the derived categories of all the usual cohomology theories.
In the paper On the geometric theory of local MV-algebras (joint with A. C. Russo), we investigate the geometric theory of local MV-algebras and its quotients axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We show that, whilst the theory of local MV-algebras is not of presheaf type, each of these quotients is a theory of presheaf type which is Morita-equivalent to an expansion of the theory of lattice-ordered abelian groups. Di Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the quotient axiomatizing the local MV-algebras in Chang's variety, that is, the perfect MV-algebras. We establish along the way a number of results of independent interest, including a constructive treatment of the radical for MV-algebras in a fixed proper variety of MV-algebras and a representation theorem for the finitely presentable algebras in such a variety as finite products of local MV-algebras.
In the paper On the depedent product in toposes (joint with Riccardo Zanfa) we give an explicit construction of the dependent product in an elementary topos, and a site-theoretic description for it in the case of a Grothendieck topos.
In the work Denseness conditions, morphisms and equivalences of toposes
we systematically investigate morphisms and equivalences of toposes from
multiple points of view. We establish a dual adjunction between
morphisms and comorphisms of sites, introduce the notion of weak
morphism of toposes and characterize the functors which induce such
morphisms. In particular, we examine continuous comorphism of sites and
show that this class of comorphisms notably includes all fibrations as
well as morphisms of fibrations. We also establish a characterization
theorem for essential geometric morphisms and locally connected
morphisms in terms of continuous functors, and a relative version of the
comprehensive factorization of a functor.
Then we prove a general theorem providing necessary and sufficient
explicit conditions for a morphism of sites to induce an equivalence of
toposes. This stems from a detailed analysis of arrows in Grothendieck
toposes and denseness conditions, which yields results of independent
interest. We also derive site characterizations of the property of a
geometric morphism to be an inclusion (resp. a surjection,
hyperconnected, localic), as well as site-level descriptions of the
surjection-inclusion and hyperconnected-localic factorizations of a
geometric morphism.
In the paper The over-topos at a model (joint with Axel Osmond), with a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. Lastly, we investigate the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes.
In the work Relative topos theory via stacks (joint with Riccardo Zanfa), we introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site (C,J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a 'petit' topos associated with a 'gros' topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.
Laurent Lafforgue has advocated in this document the importance of investigating the technique 'toposes as bridges' in connection to the Langlands programme.
A working group has formed at the Ecole Polytechnique to study some of my papers