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. In mathematics, a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

In the school of Grothendieck

During the latter part of the 1950s, the foundations of algebraic geometry 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. Millennia: 1st millennium - 2nd millennium - 3rd millennium Events and trends Technology United States tests the first fusion bomb. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack 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 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. In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name projective geometry was a stepping stone from analytic geometry to algebraic geometry. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... Several specialized usages of the terms compact and compactness exist. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...

The question of points was close to resolution by 1950; 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. 1950 was a common year starting on Sunday (link will take you to calendar). ... In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ... In category theory, a functor is a special type of mapping between categories. ...

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, 'descent' is part of the theory of comonads; here we can see the 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. 1960 was a leap year starting on Friday (link will take you to calendar). ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ... In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ...

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 can perhaps be traced back to the lattice concept. It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves of sets. In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... 1957 was a common year starting on Tuesday (link will take you to calendar). ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

This definition of a topos was eventually given, 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. 1962 was a common year starting on Monday (link will take you to calendar). ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of general cohomology theories. ... In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of general cohomology theories. ...

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 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. You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... 1964 was a leap year starting on Wednesday (link will take you to calendar). ... 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. ... In mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. ...

From pure category theory to categorical logic

The current definition of topos goes back to William Lawvere. 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 disciplines. In mathematics, a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. ... Francis William Lawvere is a mathematician who is known for his work in category theory and the philosophy of mathematics. ...

Lawvere'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. In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. Introductory example As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions...

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. In fact in terms of the space associated with a sheaf it is attractively described as the union of disjoint copies of each open set U of X. This maps to X by an obvious local homeomorphism: it looks like a stack of all the open sets of X projecting down. The stalk for x in X has a point for each U containing x; so that this sheaf looks like the graph of the membership relation. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

Lawvere 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'. In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ...

Once the idea of a connection with logic was formulated, there were several development 'testing' the new theory:

• models of set theory showing the independence of the continuum hypothesis
• recognition of the connection with intuitionistic logic's idea of the existential quantifier
• combining these, discussion of the intuitionistic theory of real numbers, by sheaf models.

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something, or at least one relevant thing. ...

Position of topos theory

There was some irony that in the pushing through of David Hilbert's long-range programme a natural home for intuitionistic logic's central ideas was found: Hilbert had detested, not even cordially, the school of Brouwer. Existence as 'local' existence in the sheaf-theoretic sense 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. David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...

The later work on etale 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. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

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 Lambek and Scott. What results is essentially an intuitionistic (i.e. constructivist) 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 needs that to be upgraded to a Boolean algebra, a return to the case of two Boolean truth-values. In that book, the talk is about constructivist 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. In linguistics, syntax is the study of the rules, or patterned relations, that govern the way the words in a sentence come together. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ... Wikibooks has more about this subject: Wikiversity Riverside Graphics Lab Open Directory Project: Computer Science Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: Computer science | Academic disciplines ...

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. Pointless topology is an approach to topology which avoids the mentioning of points. ... In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... In computer science, denotational semantics is one of the approaches to formalize the semantics of computer programs. ...

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. Granted the general view of Saunders Mac Lane about ubiquity of concepts, this gives them a definite status. As a 'killer application' one falls back on etale cohomology. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into sets called types. ... Saunders Mac Lane (born 4 August 1909) is a US mathematician. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...