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For topoi in literary theory, see Literary topos. Topos (literally "a place" pl topoi) referred in the context of classical Greek Rhetoric to a standardised method of constructing For topoi in rhetorical invention, see Inventio. Inventio is the system or method used for the discovery of arguments in Western Rhetoric and comes from the Latin word meaning "invention" or

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. For a discussion of the history of topos theory, see the article Background and genesis of topos theory. This page gives some very general background to the mathematical idea of Topos.

Contents

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. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany 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. 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. A theorem of Giraud states that the following are equivalent:

A category with these properties is called a "(Grothendieck) topos". Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. In Category theory, a branch of Mathematics, a V-valued presheaf F on a category C is a functor FC^\mathrm{op}\to\mathbf{V}

Giraud's axioms

Giraud's axioms for a category C are:

The last axiom needs the most explanation. If X is an object of C, an equivalence relation R on X is a map R→X×X in C such that all the maps Hom(Y,R)→Hom(Y,X)×Hom(Y,X) are equivalence relations of sets. Since C has colimits we may form the coequalizer of the two maps R→X; call this X/R. The equivalence relation is effective if the canonical map

R \to X \times_{X/R} X \,\!

is an isomorphism.

Examples

Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.

The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point.

More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number 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 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 . .

Counterexamples

Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. In Mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points. Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician.

Geometric morphisms

If X and Y are topoi, a geometric morphism u:X→Y is a pair of adjoint functors (u∗,u∗) such that u∗ preserves finite limits. Note that u∗ automatically preserves colimits by virtue of having a right adjoint.

By Freyd's adjoint functor theorem, to give a geometric morphism X → Y is to give a functor Y → X that preserves finite limits and small colimits.

If X and Y are topological spaces and u is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi.

Points of topoi

A point of a topos X is a geometric morphism from the topos of sets to X.

If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk Fx has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point.

Ringed topoi

A ringed topos is a pair (X,R), where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through for ringed topoi. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on The category of R-module objects in X is an abelian category with enough injectives. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist A more useful abelian category is the subcategory of quasi-coherent R-modules: these are R-modules that admit a presentation.

Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks. In Algebraic geometry, a branch of Mathematics, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and

Homotopy theory of topoi

Michael Artin and Barry Mazur associated to any topos a pro-simplicial set. Michael Artin (born 1934 is an American Mathematician and a professor at MIT, known for his contributions to Algebraic Barry Charles Mazur (born December 19, 1937) is a professor of mathematics at Harvard University. In Mathematics, a pro-simplicial set is an Inverse system of Simplicial sets A pro-simplicial set is called pro-finite if each term of the Using this inverse system of simplicial sets one may sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory. In Mathematics, an inverse system in a category C is a Functor from a small cofiltered category I to C. In Mathematics, an inverse system in a category C is a Functor from a small cofiltered category I to C.

The pro-simplicial set associated to the etale topos of a scheme is a pro-finite simplicial set. In Mathematics, a pro-simplicial set is an Inverse system of Simplicial sets A pro-simplicial set is called pro-finite if each term of the Its study is called étale homotopy theory.

Elementary toposes (toposes 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). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function More recent work in category theory allows this foundation to be generalized using toposes; each topos completely defines its own mathematical framework. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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 alternative toposes. A standard formulation of the axiom of choice makes sense in any topos, and there are toposes in which it is invalid. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. Constructivists will be interested to work in a topos without the law of excluded middle. In the Philosophy of mathematics This article uses forms of logical notation For a concise description of the symbols used in this notation see Table of logic symbols. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

It is also possible to encode an algebraic theory, such as the theory of groups, as a topos. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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 Category theory, a branch of Mathematics, 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:

From this one can derive that

In many applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.

Explanation

A topos as defined above can be understood as a cartesian closed category for which the notion of subobject of an object has an elementary or first-order definition. 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 First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science This notion, as a natural categorical abstraction of the notions of subset of a set, subgroup of a group, and more generally subalgebra of any algebraic structure, predates the notion of topos. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, It is definable in any category, not just toposes, in second-order language, i. In Logic and Mathematics second-order logic is an extension of First-order logic, which itself is an extension of Propositional logic. e. in terms of classes of morphisms instead of individual morphisms, as follows. Given two monics m, n from respectively Y and Z to X, we say that m ≤ n when there exists a morphism p: Y → Z for which np = m, inducing a preorder on monics to X. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions When m ≤ n and n ≤ m we say that m and n are equivalent. The subobjects of X are the resulting equivalence classes of the monics to it.

In a topos "subobject" becomes, at least implicitly, a first-order notion, as follows.

As noted above, a topos is a category C having all finite limits and hence in particular the empty limit or final object 1. It is then natural to treat morphisms of the form x: 1 → X as elements x ∈ X. Morphisms f: X → Y thus correspond to functions mapping each element x ∈ X to the element fx ∈ Y, with application realized by composition.

One might then think to define a subobject of X as an equivalence class of monics m: X' → X having the same image or range {mx| x ∈ X' }. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, the range of a function is the set of all "output" values produced by that function The catch is that two or more morphisms may correspond to the same function, that is, we cannot assume that C is concrete in the sense that the functor C(1,-): C → Set is faithful. For example the category Grph of graphs and their associated homomorphisms is a topos whose final object 1 is the graph with one vertex and one edge (a self-loop), but is not concrete because the elements 1 → G of a graph G correspond only to the self-loops and not the other edges, nor the vertices without self-loops. A multigraph or pseudograph is a graph which is permitted to have Multiple edges, (also called "parallel edges" that is edges that have the In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector Whereas the second-order definition makes G and its set of self-loops (with their vertices) distinct subobjects of G (unless every edge is, and every vertex has, a self-loop), this image-based one does not. This can be addressed for the graph example and related examples via the Yoneda Lemma as described in the Examples section below, but this then ceases to be first-order. In Mathematics, specifically in Category theory, the Yoneda lemma is an abstract result on Functors of the type morphisms into a fixed object. Toposes provide a more abstract, general, and first-order solution.

Figure 1. m as a pullback of the generic subobject t along f.
Figure 1. m as a pullback of the generic subobject t along f.

As noted above a topos C has a subobject classifier Ω, namely an object of C with an element t ∈ Ω, the generic subobject of C, having the property that every monic m: X' → X arises as a pullback of the generic subobject along a unique morphism f: X → Ω, as per Figure 1. In Category theory, a subobject classifier is a special object &Omega of a category intuitively the Subobjects of an object X correspond to the morphisms In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. Now the pullback of a monic is a monic, and all elements including t are monics since there is only one morphism to 1 from any given object, whence the pullback of t along f: X → Ω is a monic. The monics to X are therefore in bijection with the pullbacks of t along morphisms from X to Ω. The latter morphisms partition the monics into equivalence classes each determined by a morphism f: X → Ω, the characteristic morphism of that class, which we take to be the subobject of X characterized or named by f.

All this applies to any topos, whether or not concrete. In the concrete case, namely C(1,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions. Here the monics m: X' → X are exactly the injections (one-one functions) from X' to X, and those with a given image {mx| x ∈ X' } constitute the subobject of X corresponding to the morphism f: X → Ω for which f−1(t) is that image. The monics of a subobject will in general have many domains, all of which however will be in bijection with each other.

To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics to X as had previously been defined explicitly by the second-order notion of subobject for any category. The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobject classifier Ω, leaving the notion of subobject of X as an implicit consequence characterized (and hence namable) by its associated morphism f: X → Ω.

Further examples

If C is a small category, then the functor category SetC (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos. A graph consists of two sets, an edge set and a vertex set, and two functions s,t between those sets, assigning to every edge e its source s(e) and target t(e). Grph is thus equivalent to the functor category SetC, where C is the category with two objects E and V and two morphisms s,t: E → V giving respectively the source and target of each edge. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are

The categories of finite sets, of finite G-sets (actions of a group G on a finite set), and of finite graphs are also toposes.

The Yoneda Lemma asserts that Cop embeds in SetC as a full subcategory. In Mathematics, specifically in Category theory, the Yoneda lemma is an abstract result on Functors of the type morphisms into a fixed object. In the graph example the embedding represents Cop as the subcategory of SetC whose two objects are V' as the one-vertex no-edge graph and E' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from V' to E' (both as natural transformations). The natural transformations from V' to an arbitrary graph (functor) G constitute the vertices of G while those from E' to G constitute its edges. Although SetC, which we can identify with Grph, is not made concrete by either V' or E' alone, the functor U: Grph → Set2 sending object G to the pair of sets (Grph(V',G), Grph(E',G)) and morphism h: G → H to the pair of functions (Grph(V',h), Grph(E',h)) is faithful. That is, a morphism of graphs can be understood as a pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of generalized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of toposes by allowing an object to have multiple underlying sets, that is, to be multisorted.

References

Some gentle papers:

The following texts are easy-paced introductions to toposes and the basics of category theory. They should be suitable for those knowing little mathematical logic and set theory, even non-mathematicians.

Grothendieck foundational work on toposes:

The following monographs include an introduction to some or all of topos theory, but do not cater primarily to beginning students. Listed in (perceived) order of increasing difficulty.

Reference works for experts, less suitable for first introduction:

Books that target special applications of topos theory:

Online encyclopedias:

See also

PlanetMath is a free, collaborative online Mathematics Encyclopedia. This page gives some very general background to the mathematical idea of Topos. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a Logical system and a Set theory

Dictionary

topos

-noun

  1. A literary theme or motif; a rhetorical convention or formula.
  2. (mathematics) A certain mathematical structure found in category theory.
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