Background and genesis of topos theory
Encyclopedia
This page gives some very general background to the mathematical idea of topos
. This is an aspect of category theory
, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory.
were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures
were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of étale cohomology
.
With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems, for a long time. The first was to do with its points: back in the days of projective geometry
it was clear that the absence of 'enough' points on an algebraic variety
was a barrier to having a good geometric theory (in which it was somewhat like a compact
manifold
). There was also the difficulty, that was clear as soon as topology
took form in the first half of the twentieth century, that the topology of algebraic varieties had 'too few' open sets.
The question of points was close to resolution by 1950; Alexander Grothendieck
took a sweeping step (appealing to the Yoneda lemma
) that disposed of it — naturally at a cost, that every variety or more general scheme
should become a functor
. It wasn't possible to add open sets, though. The way forward was otherwise.
The topos definition first appeared somewhat obliquely, in or about 1960. General problems of so-called 'descent
' in algebraic geometry were considered, at the same period when the fundamental group
was generalised to the algebraic geometry setting (as a pro-finite group
). In the light of later work (c. 1970), 'descent' is part of the theory of comonads; here we can see one way in which the Grothendieck school bifurcates in its approach from the 'pure' category theorists, a theme that is important for the understanding of how the topos concept was later treated.
There was perhaps a more direct route available: the abelian category
concept had been introduced by Grothendieck in his foundational work on homological algebra
, to unify categories of sheaves
of abelian groups, and of modules
. An abelian category is supposed to be closed under certain category-theoretic operations — by using this kind of definition one can focus entirely on structure
, saying nothing at all about the nature of the objects involved. This type of definition traces back, in one line, to the lattice
concept of the 1930s. It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves
of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work (the Tohoku paper).
Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier
(see Verdier's Bourbaki seminar Analysis Situs). The characterisation was by means of categories 'with enough colimits', and applied to what is now called a Grothendieck topos. The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology
.
The idea of a Grothendieck topology (also known as a site) has been characterised by John Tate
as a bold pun on the two senses of Riemann surface
. Technically speaking it enabled the construction of the sought-after étale cohomology (as well as other refined theories such as flat cohomology and crystalline cohomology
). At this point — about 1964 — the developments powered by algebraic geometry had largely run their course. The 'open set' discussion had effectively been summed up in the conclusion that varieties had a rich enough site of open sets in unramified
covers of their (ordinary) Zariski-open sets
.
goes back to William Lawvere
and Myles Tierney. While the timing follows closely on from that described above, as a matter of history, the attitude is different, and the definition is more inclusive. That is, there are examples of toposes that are not Grothendieck topos. What is more, these may be of interest for a number of logical disciplines.
Lawvere and Tierney's definition picks out the central role in topos theory of the sub-object classifier. In the usual category of sets, this is the two-element set of Boolean truth-values, true and false. It is almost tautologous to say that the subsets of a given set X are the same as (just as good as) the functions on X to any such given two-element set: fix the 'first' element and make a subset Y correspond to the function sending Y there and its complement in X to the other element.
Now sub-object classifiers can be found in sheaf
theory. Still tautologously, though certainly more abstractly, for a topological space
X there is a direct description of a sheaf on X that plays the role with respect to all sheaves of sets on X. Its set of sections over an open set U of X is just the set of open subsets of U. The space associated with a sheaf
for it is more difficult to describe.
Lawvere and Tierney therefore formulated axioms for a topos that assumed a sub-object classifier, and some limit conditions (to make a cartesian-closed category, at least). For a while this notion of topos was called 'elementary topos'.
Once the idea of a connection with logic was formulated, there were several developments 'testing' the new theory:
's long-range programme a natural home for intuitionistic logic
's central ideas was found: Hilbert had detested the school of L. E. J. Brouwer. Existence as 'local' existence in the sheaf-theoretic sense, now going by the name of Kripke–Joyal semantics, is a good match. On the other hand Brouwer's long efforts on 'species', as he called the intuitionistic theory of reals, are presumably in some way subsumed and deprived of status beyond the historical. There is a theory of the real numbers in each topos, and so no one master intuitionist theory.
The later work on étale cohomology
has tended to suggest that the full, general topos theory isn't required. On the other hand, other sites are used, and the Grothendieck topos has taken its place within homological algebra.
The Lawvere programme was to write higher-order logic in terms of category theory. That this can be done cleanly is shown by the book treatment by Joachim Lambek
and P. S. Scott. What results is essentially an intuitionistic (i.e. constructive logic) theory, its content being clarified by the existence of a free topos. That is a set theory, in a broad sense, but also something belonging to the realm of pure syntax
. The structure on its sub-object classifier is that of a Heyting algebra
. To get a more classical set theory one can look at toposes in which it is moreover a Boolean algebra, or specialising even further, at those with just two truth-values. In that book, the talk is about constructive mathematics
; but in fact this can be read as foundational computer science
(which is not mentioned). If one wants to discuss set-theoretic operations, such as the formation of the image (range) of a function, a topos is guaranteed to be able to express this, entirely constructively.
It also produced a more accessible spin-off in pointless topology
, where the locale concept isolates some of more accessible insights found by treating topos as a significant development of topological space. The slogan is 'points come later': this brings discussion full circle on this page. The point of view is written up in Peter Johnstone's Stone Spaces, which has been called by a leader in the field of computer science 'a treatise on extensionality
'. The extensional is treated in mathematics as ambient - it is not something about which mathematicians really expect to have a theory. Perhaps this is why topos theory has been treated as an oddity; it goes beyond what the traditionally geometric way of thinking allows. The needs of thoroughly intensional theories such as untyped lambda calculus
have been met in denotational semantics
. Topos theory has long looked like a possible 'master theory' in this area.
The subsequent developments associated with logic are more interdisciplinary. They include examples drawing on homotopy theory (classifying toposes). They involve links between category theory and mathematical logic, and also (as a high-level, organisational discussion) between category theory and theoretical computer science based on type theory
. Granted the general view of Saunders Mac Lane
about ubiquity of concepts, this gives them a definite status. A 'killer application' is étale cohomology
.
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
. This is an aspect of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory.
In the school of Grothendieck
During the latter part of the 1950s, the foundations of algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
.
With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems, for a long time. The first was to do with its points: back in the days of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
it was clear that the absence of 'enough' points on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
was a barrier to having a good geometric theory (in which it was somewhat like a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
). There was also the difficulty, that was clear as soon as topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
took form in the first half of the twentieth century, that the topology of algebraic varieties had 'too few' open sets.
The question of points was close to resolution by 1950; Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
took a sweeping step (appealing to the Yoneda lemma
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
) that disposed of it — naturally at a cost, that every variety or more general scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
should become a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
. It wasn't possible to add open sets, though. The way forward was otherwise.
The topos definition first appeared somewhat obliquely, in or about 1960. General problems of so-called 'descent
Descent (category theory)
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
' in algebraic geometry were considered, at the same period when the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
was generalised to the algebraic geometry setting (as a pro-finite group
Pro-finite group
In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.- Definition :...
). In the light of later work (c. 1970), 'descent' is part of the theory of comonads; here we can see one way in which the Grothendieck school bifurcates in its approach from the 'pure' category theorists, a theme that is important for the understanding of how the topos concept was later treated.
There was perhaps a more direct route available: the abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
concept had been introduced by Grothendieck in his foundational work on homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
, to unify categories of sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of abelian groups, and of modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
. An abelian category is supposed to be closed under certain category-theoretic operations — by using this kind of definition one can focus entirely on structure
Structure (category theory)
In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications...
, saying nothing at all about the nature of the objects involved. This type of definition traces back, in one line, to the lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
concept of the 1930s. It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work (the Tohoku paper).
Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier
Jean-Louis Verdier
Jean-Louis Verdier was a French mathematician who worked, under the guidance of Alexander Grothendieck, on derived categories and Verdier duality...
(see Verdier's Bourbaki seminar Analysis Situs). The characterisation was by means of categories 'with enough colimits', and applied to what is now called a Grothendieck topos. The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
.
The idea of a Grothendieck topology (also known as a site) has been characterised by John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
as a bold pun on the two senses of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
. Technically speaking it enabled the construction of the sought-after étale cohomology (as well as other refined theories such as flat cohomology and crystalline cohomology
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field....
). At this point — about 1964 — the developments powered by algebraic geometry had largely run their course. The 'open set' discussion had effectively been summed up in the conclusion that varieties had a rich enough site of open sets in unramified
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
covers of their (ordinary) Zariski-open sets
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
.
From pure category theory to categorical logic
The current definition of toposTopos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
goes back to William Lawvere
William Lawvere
Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...
and Myles Tierney. While the timing follows closely on from that described above, as a matter of history, the attitude is different, and the definition is more inclusive. That is, there are examples of toposes that are not Grothendieck topos. What is more, these may be of interest for a number of logical disciplines.
Lawvere and Tierney's definition picks out the central role in topos theory of the sub-object classifier. In the usual category of sets, this is the two-element set of Boolean truth-values, true and false. It is almost tautologous to say that the subsets of a given set X are the same as (just as good as) the functions on X to any such given two-element set: fix the 'first' element and make a subset Y correspond to the function sending Y there and its complement in X to the other element.
Now sub-object classifiers can be found in sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
theory. Still tautologously, though certainly more abstractly, for a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X there is a direct description of a sheaf on X that plays the role with respect to all sheaves of sets on X. Its set of sections over an open set U of X is just the set of open subsets of U. The space associated with a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
for it is more difficult to describe.
Lawvere and Tierney therefore formulated axioms for a topos that assumed a sub-object classifier, and some limit conditions (to make a cartesian-closed category, at least). For a while this notion of topos was called 'elementary topos'.
Once the idea of a connection with logic was formulated, there were several developments 'testing' the new theory:
- models of set theory corresponding to proofs of the independence of the axiom of choice and continuum hypothesisContinuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
by Cohen's method of forcingForcing (mathematics)In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
. - recognition of the connection with Kripke semanticsKripke semanticsKripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
, the intuitionisticIntuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
existential quantifier and intuitionistic type theoryIntuitionistic type theoryIntuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher,...
. - combining these, discussion of the intuitionistic theory of real numbers, by sheaf models.
Position of topos theory
There was some irony that in the pushing through of David HilbertDavid Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
's long-range programme a natural home for intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
's central ideas was found: Hilbert had detested the school of L. E. J. Brouwer. Existence as 'local' existence in the sheaf-theoretic sense, now going by the name of Kripke–Joyal semantics, is a good match. On the other hand Brouwer's long efforts on 'species', as he called the intuitionistic theory of reals, are presumably in some way subsumed and deprived of status beyond the historical. There is a theory of the real numbers in each topos, and so no one master intuitionist theory.
The later work on étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
has tended to suggest that the full, general topos theory isn't required. On the other hand, other sites are used, and the Grothendieck topos has taken its place within homological algebra.
The Lawvere programme was to write higher-order logic in terms of category theory. That this can be done cleanly is shown by the book treatment by Joachim Lambek
Joachim Lambek
Joachim Lambek is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Julius Zassenhaus as advisor. He is called Jim by his friends.- Scholarly work :...
and P. S. Scott. What results is essentially an intuitionistic (i.e. constructive logic) theory, its content being clarified by the existence of a free topos. That is a set theory, in a broad sense, but also something belonging to the realm of pure syntax
Syntax
In linguistics, syntax is the study of the principles and rules for constructing phrases and sentences in natural languages....
. The structure on its sub-object classifier is that of a Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
. To get a more classical set theory one can look at toposes in which it is moreover a Boolean algebra, or specialising even further, at those with just two truth-values. In that book, the talk is about constructive mathematics
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
; but in fact this can be read as foundational computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
(which is not mentioned). If one wants to discuss set-theoretic operations, such as the formation of the image (range) of a function, a topos is guaranteed to be able to express this, entirely constructively.
It also produced a more accessible spin-off in pointless topology
Pointless topology
In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
, where the locale concept isolates some of more accessible insights found by treating topos as a significant development of topological space. The slogan is 'points come later': this brings discussion full circle on this page. The point of view is written up in Peter Johnstone's Stone Spaces, which has been called by a leader in the field of computer science 'a treatise on extensionality
Extensionality
In logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties...
'. The extensional is treated in mathematics as ambient - it is not something about which mathematicians really expect to have a theory. Perhaps this is why topos theory has been treated as an oddity; it goes beyond what the traditionally geometric way of thinking allows. The needs of thoroughly intensional theories such as untyped lambda calculus
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
have been met in denotational semantics
Denotational semantics
In computer science, denotational semantics is an approach to formalizing the meanings of programming languages by constructing mathematical objects which describe the meanings of expressions from the languages...
. Topos theory has long looked like a possible 'master theory' in this area.
Summary
The topos concept arose in algebraic geometry, as a consequence of combining the concept of sheaf and closure under categorical operations. It plays a certain definite role in cohomology theories.The subsequent developments associated with logic are more interdisciplinary. They include examples drawing on homotopy theory (classifying toposes). They involve links between category theory and mathematical logic, and also (as a high-level, organisational discussion) between category theory and theoretical computer science based on type theory
Type theory
In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
. Granted the general view of Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
about ubiquity of concepts, this gives them a definite status. A 'killer application' is étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
.