AskDefine | Define topos

Dictionary Definition

topos n : a traditional theme or motif or literary convention; "James Joyce uses the topos of the Wandering Jew in his Ulysses" [also: topoi (pl)]

User Contributed Dictionary

English

Etymology

From τόπος ‘place’. Compare topic.

Pronunciation

  • /ˈtɒpɒs/

Noun

  1. A literary theme or motif; a rhetorical convention or formula.
    • 2003, Roy Porter, Flesh in the Age of Reason (Penguin 2004, p. 239)
      The ritual of weighing the soul was an iconographic topos familiar to Christianity from the ceremony of the weighing of sins at the Last Judgement.
  2. A certain mathematical structure found in category theory.

Translations

literary theme
  • Dutch: topos
mathematical structure
  • Dutch: topos

Dutch

Pronunciation

  • ˈtopɔs|lang=nl

Noun

  1. Topos, literary theme.
  2. Topos, mathematical structure.

Extensive Definition

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:
  • There is a small category D and an inclusion C \hookrightarrow 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". 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.

Giraud's axioms

Giraud's axioms for a category C are:
  • C has a small set of generators, and admits all small 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 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 \,\!
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 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. 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.

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). More recent work in category theory allows this foundation to be generalized using toposes; 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 alternative toposes. A standard formulation of the axiom of choice makes sense in any topos, and there are toposes 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:
  • All limits taken over finite index categories exist.
  • Every object has a power object.
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. 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. It is definable in any category, not just toposes, in second-order language, i.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. 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 . 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. 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. Toposes provide a more abstract, general, and first-order solution.
topos in Arabic: توبوس
topos in German: Topos (Mathematik)
topos in Spanish: Topos
topos in French: Topos (mathématiques)
topos in Dutch: Topos
topos in Japanese: トポス (数学)
topos in Russian: Теория топосов
topos in Swedish: Toposteori
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