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This article gives some very general background to the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
idea of
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
. This is an aspect of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, 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 geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. T ...
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 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 conject ...
. 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 In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
it was clear that the absence of 'enough' points on an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
was a barrier to having a good geometric theory (in which it was somewhat like a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
). There was also the difficulty, that was clear as soon as
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
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 (invoking the
Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important result in category theory. It 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 (view ...
) that disposed of it—naturally at a cost, that every variety or more general '' scheme'' should become a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
. 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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
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
comonad In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is an ...
s; 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 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 prototypical example of an abelian category is the category of a ...
concept had been introduced by Grothendieck in his foundational work on
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
, to unify categories of sheaves of abelian groups, and of
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
. 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 can be traced back, in one line, to the lattice concept of the 1930s. It was a possible question to ask, around 1957, for a 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 ''Tôhoku'' paper). Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier (see Verdier's
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
seminar ''Analysis Situs''). The characterisation was by means of categories 'with enough
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions suc ...
s', and applied to what is now called a Grothendieck topos. The theory was rounded out by establishing that a Grothendieck topos was a category of sheaves, where now the word ''sheaf'' had acquired an extended meaning, since it involved a
Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
. 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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versi ...
. 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 In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . ...
). 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.


From pure category theory to categorical logic

The current definition of
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
goes back to
William Lawvere Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Lawvere studied continuum mechanics as an undergraduate with Clifford Truesd ...
and
Myles Tierney Myles Tierney (September 1937 – 5 October 2017) was an American mathematician and Professor at Rutgers University who founded the theory of elementary toposes with William Lawvere. Tierney obtained his B.A. from Brown University in 1959 and h ...
. 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 a Grothendieck topos. What is more, these may be of interest for a number of
logical Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
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-value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...
s, 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 may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper s ...
theory. Still tautologously, though certainly more abstractly, for a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''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: * models of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
corresponding to proofs of the independence of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
and
continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
by
Paul Cohen Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
's method of forcing. * recognition of the connection with
Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...
, the intuitionistic
existential quantifier In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
and
intuitionistic type theory Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and p ...
. * 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 Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
's long-range programme a natural home for
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems o ...
's central ideas was found: Hilbert had detested the school of
L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and c ...
. 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 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 conject ...
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 mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expres ...
in terms of category theory. That this can be done cleanly is shown by the book treatment by
Joachim Lambek Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus as a ...
and P. J. Scott. What results is essentially an intuitionistic (i.e.
constructive logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems o ...
) 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 In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency ...
. The structure on its sub-object classifier is that of a
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
. To get a more classical set theory one can look at toposes in which it is moreover a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
, or specialising even further, at those with just two truth-values. In that book, the talk is about
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
; 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 In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this appr ...
, where the '' locale'' concept isolates some 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 In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
'. 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 (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation th ...
have been met in
denotational semantics In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called ''denotations' ...
. 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. A 'killer application' is
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 conject ...
. The subsequent developments associated with logic are more interdisciplinary. They include examples drawing on
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
( 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 In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a found ...
. Granted the general view of
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
about ''ubiquity'' of concepts, this gives them a definite status. The use of toposes as unifying bridges in mathematics has been pioneered by Olivia Caramello in her 2017 book.


References

*http://plato.stanford.edu/entries/category-theory/ {{DEFAULTSORT:Background And Genesis Of Topos Theory Topos theory History of mathematics