In mathematics, a topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... 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. ... This page gives some very general background to the mathematical idea of topos. ...

Grothendieck topoi (topoi in geometry)

Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on that space. This idea was expounded by Alexander Grothendieck by introducing the notion of a topos. The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest single success of this programmatic idea to date has been the introduction of the étale topos of a scheme. Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

Equivalent formulations

Let C be a category. The following are equivalent:

• There is a category D and an inclusion C Presh(D) that admits a left adjoint.
• C is the category of sheaves on a Grothendieck site.
• C satisfies Giraud's axioms, below.

A category with these properties is called a "(Grothendieck) topos". The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... 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. ...

Giraud's axioms

Giraud's axioms for a category C are:

• C has a small set of generators, and admits arbitrary colimits. Furthermore, colimits commute with base change.
• Sums in C are disjoint. In other words, the fiber product of X and Y over their sum is the initial object in C.
• All equivalence relations in C are effective.

The last axiom needs the most explanation. If X is an object of C, an equivalence relation R on X is a map in C such that all the maps are equivalence relations of sets. Since C has colimits we may form the coequalizer of the two maps ; call this X/R. The equivalence relation is effective if the canonical map 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 category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...

is an isomorphism.

Ringed topoi

A ringed topos is a pair (X,R), where X is a topos and R is a ring object in X. Most of the constructions of ringed spaces go through for ringed topoi. The category of R-module objects in X is an abelian category with enough injectives. A more useful abelian category is the subcategory of quasi-coherent R-modules: these are R-modules that admit a presentation. In mathematics, a ringed space is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... 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 nice properties. ...

Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks. In mathematics, an algebraic stack in algebraic geometry is a special case of the concept of a stack, which is useful for working on moduli questions. ...

Homotopy theory of topoi

Michael Artin and Barry Mazur associated to any topos a pro-simplicial set. This led in particular to their definition of étale homotopy theory. Michael Artin is an American mathematician, known for his contributions to algebraic geometry. ... Barry Mazur (born December 19, 1937) is a professor of mathematics at Harvard University. ...

Elementary topoi (topoi in logic)

Introduction

A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions which map between sets.) More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic mathematics. But one could instead choose to work with many alternate topoi. A topos exists in which the axiom of choice is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets. Another important example of a topos (and historically the first) is the category of all sheaves of sets on a given topological space. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Partial plot of a function f. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

It is also possible to encode a logical theory, such as the theory of all groups, in a topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure. In mathematical logic, a theory is usually defined as a set of first-order sentences (closed first-order formulas). ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Formal definition

When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise, if not illuminating:

A topos is a category which has the following two properties: Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

• All limits taken over finite index categories exist.
• Every object has a power object.

From this one can derive that 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, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. ...

• All colimits taken over finite index categories exist.
• The category has a subobject classifier.
• Any two objects have an exponential object.
• The category is cartesian closed.

In many applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what's defined and what's derived. 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 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... In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. ... 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. ...

Further examples

If C is a small category, then the functor category Set^C (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category of all directed graphs is a topos. A graph consists of two sets, an arrow set and a vertex set, and two functions between those sets, assigning to every arrow its start and end vertex. The category of graphs is thus equivalent to the functor category Set^C, where C is the category with two objects joined by two morphisms. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... A labeled graph with 6 vertices and 7 edges. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ...

The categories of finite sets, of finite G-sets and of finite directed graphs are also topoi.

References

• John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.
• Stephen Vickers: Toposes pour les nuls and Toposes pour les vraiment nuls. Available at Vickers’ website. Elementary and even more elementary introductions to toposes as generalized spaces.

The following textbooks provide easy paced first introductions (including basics of category theory). They should be suitable for students of various—even non-mathematical—disciplines: John Carlos Baez (b. ...

• F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
• F. William Lawvere and Robert Rosebrugh: Sets for Mathematics, Cambridge University Press, Cambridge, 2003. Discusses the foundations of mathematics from a categorical perspective. A book "for students who are beginning the study of advanced mathematical subjects".

The original work of Grothendieck Francis William Lawvere is a mathematician who is known for his work in category theory and the philosophy of mathematics. ...

• Grothendieck and Verdier: Théorie des topos et cohomologie étale des schémas (known as SGA4)". New York/Berlin: Springer, ??. (Lecture notes in mathematics, 269–270)

Interesting research books that are provide introductions to topos theory (or to a specific aspect of it), but which do not primarily cater to students. The given order roughly (!) reflects the difficulty of the level of exposition: Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois...

• Colin McLarty: Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1992. Includes a nice introduction of the basic notions of category theory, topos theory, and topos logic. Assumes very few prerequisites.
• Robert Goldblatt: Topoi, the Categorial Analysis of Logic. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.

This book is now out of print and the copyright has reverted to the author. It can be accessed freely on Robert Goldblatt's homepage: Topoi, the Categorical Analysis of Logic.

• John L. Bell: The development of categorical logic. http://publish.uwo.ca/~jbell/catlogprime.pdf (PDF)
• Saunders Mac Lane and Ieke Moerdijk: Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Springer, New York, 1992. More complete, and more difficult to read.
• Michael Barr and Charles Wells: Toposes, Triples and Theories, Springer, 1985. Corrected online version at http://www.cwru.edu/artsci/math/wells/pub/ttt.html. More concise than Sheaves in Geometry and Logic, but not an easy reading for the beginner.

Works which serve as a reference for experts in the field rather than as a treatment suitable for first introduction: PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...

• Francis Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves, Volume 52 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1994. The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given.
• Peter T. Johnstone: Topos Theory, L. M. S. Monographs no. 10, Academic Press, 1977. For a long time the standard compendium on topos theory. However, it has also been described as "far too hard to read, and not for the faint-hearted", as quoted by Johnstone himself.
• Peter T. Johnstone: Sketches of an Elephant: A Topos Theory Compendium, Oxford Science Publications, Oxford, 2002. Johnstone’s overwhelming compendium. As of early 2006, two of the scheduled three volumes were available.

Books that target special applications of topos theory:

• Maria Cristina Pedicchio and Walter Tholen (editors): Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Volume 97 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004. Includes many interesting special applications.