# 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

### Pronunciation

- /ˈtɒpɒs/

### Noun

- 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.

- 2003, Roy Porter,
Flesh in the Age of Reason (Penguin 2004, p. 239)
- A certain mathematical structure found in category theory.

#### Translations

literary theme

- Dutch: topos

mathematical structure

- Dutch: topos

## Dutch

### Pronunciation

- ˈtopɔs|lang=nl

# 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

- 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

- 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 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