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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. 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. 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. Girauds 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 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 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... 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. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points. 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 limits and colimits. Note that u∗ automatically preserves colimits by virtue of having a right adjoint. 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. 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. Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks. Homotopy theory of topoi Michael Artin and Barry Mazur associated to any topos a pro-simplicial set. 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. The pro-simplicial set associated to the etale topos of a scheme is a pro-finite simplicial set. Its study is called étale homotopy theory. 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 standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it 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. It is also possible to encode an algebraic theory, such as the theory of groups, as 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. 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's defined and what's derived. 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. 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 SetC, where C is the category with two objects joined by two morphisms. The categories of finite sets, of finite G-sets and of finite directed graphs are also topoi. | |||||||
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