nLab
topos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Category Theory

Contents

Idea

There are various different perspectives on the notion of topos. One is that a topos is a category that looks like a category of spaces that sit by local homeomorphisms over a given base space: all spaces that are locally modeled on a given base space.

The archetypical class of examples are sheaf toposes Sh(X)=Et(X)Sh(X) = Et(X) over a topological space XX: these are the categories of étale spaces over XX, topological spaces YY that are equipped with a local homeomorphisms YXY \to X.

When X=*X = * is the point, this is just the category Set of all sets: spaces that are modeled on the point . This is the archetypical topos itself.

What makes the notion of topos powerful is the following fact: even though the general topos contains objects that are considerably different from and possibly considerably richer than plain sets and even richer than étale spaces over a topological space, the general abstract category theoretic properties of every topos are essentially the same as those of Set. For instance in every topos all small limits and colimits exist and it is cartesian closed (even locally). This means that a large number of constructions in Set have immediate analogs internal to every topos, and the analogs of the statements about these constructions that are true in SetSet are true in every topos.

On the one hand this may be thought of as saying that toposes are very nice categories of spaces in that whatever construction on spaces one thinks of – for instance formation of quotients or of fiber products or of mapping spaces – the resulting space with the expected general abstract properties will exist in the topos. In this sense toposes are convenient categories for geometry – as in: convenient category of topological spaces, but even more convenient than that.

On the other hand, by de-emphasizing the geometric interpretation of their objects and just using their good abstract properties, this means that toposes are contexts with a powerful internal logic. The internal logic of toposes is intuitionistic higher order logic. This means that, while the law of excluded middle and the axiom of choice may fail, apart from that, every logical statement not depending on these does hold internal to every topos.

For this reason toposes are often studied as abstract contexts “in which one can do mathematics”, independently of their interpretation as categories of spaces. These two points of views on toposes, as being about geometry and about logic at the same time, is part of the richness of topos theory.

On a third hand, however, we can de-emphasize the role of the objects of the topos and instead treat the topos itself as a “generalized space” (and in particular, a categorified space). We then consider the topos Sh(X)Sh(X) as a representative of XX itself, while toposes not of this form are “honestly generalized” spaces. This point of view is supported by the fact that the assignment XSh(X)X\mapsto Sh(X) is a full embedding of (sufficiently nice) topological spaces into toposes, and that many topological properties of a space XX can be detected at the level of Sh(X)Sh(X). (This is even more true once we pass to (∞,1)-toposes.)

From this point of view, the objects of a topos (regarded as a category) should be thought of instead as sheaves on that topos (regarded as a generalized space). And just as sheaves on a topological space can be identified with local homeomorphisms over it, such “sheaves on a topos” (i.e. objects of the topos qua category) can be identified with other toposes that sit over the given topos via a local homeomorphism of toposes.

Finally, mixing this point of view with the second one, we can regard toposes over a given topos EE instead as “toposes in the EE-world of mathematics.” For this reason, the theory of toposes over a given base is formally quite similar to that of arbitrary toposes. And coming full circle, this fact allows the use of “base change arguments” as a very useful technical tool, even if our interest is only in one or two particular toposes qua categories.

‘What a topos is like:’

(i) ‘A topos is a category of sheaves on a site’

(ii) ‘A topos is a category with finite limits and power-objects’

(iii) ‘A topos is (the embodiment of) an intuitionistic higher-order theory’

(iv) ‘A topos is (the extensional essence of) a first-order (infinitary) geometric theory’

(v) ‘A topos is a totally cocomplete object in the meta-2-category CART of cartesian (i.e. , finitely complete) categories’

(vi) ‘A topos is a generalized space’

(vii) ‘A topos is a semantics for intuitionistic formal systems’

(viii) ‘A topos is a Morita equivalence class of continuous groupoids’

(ix) ‘A topos is the category of maps of a power allegory’

(x) ‘A topos is a category whose canonical indexing over itself is complete and well-powered’

(xi) ‘A topos is the spatial manifestation of a giraud frame’

(xii) ‘A topos is a setting for synthetic differential geometry’

(xiii) ‘A topos is a setting for synthetic domain theory’,

And so on. But the important thing about the elephant is that ‘however you approach it, it is still the same animal’. Elephant

Definitions

The general notion of topos is that of

A specialization of this which is important enough that much of the literature implicitly takes it to be the general definition is the notion of

This is the notion relevant for applications in geometry and geometric logic, whereas the notion of elementary toposes is relevant for more general applications in logic.

For standard notions of mathematics to be available inside a given topos one typically at least needs a natural numbers object. Its existence is guaranteed by the axioms of a sheaf topos, but not by the more general axioms of an elementary topos. Adding the existence of a natural numbers object to the axioms of an elementary topos yields the notion of a

Elementary toposes

A quick formal definition is that an elementary topos is a category which

  1. has finite limits,

  2. is cartesian closed, and

  3. has a subobject classifier.

There are alternative ways to state the definition; for instance,

  1. has finite limits and
  2. has power objects.

In a way, however, these concise definitions can be misleading, because a topos has a great deal of other structure, which plays a very important role but just happens to follow automatically from these basic axioms. Most importantly, an elementary topos is all of the following:

The last two imply that it has an internal logic that resembles ordinary mathematical reasoning, and the presence of exponentials and power objects means that this logic is higher order.

Grothendieck/sheaf toposes

The above is the definition of an elementary topos. We also have the (historically earlier) notion of Grothendieck topos: a Grothendieck topos is a topos that is neither too small nor too large, in that it is:

Equivalently, a Grothendieck topos is any category equivalent to the category of sheaves on some small site.

WW-toposes

There is a further elementary property of Set that might have gone into the definition of elementary topos but historically did not: the existence of a natural numbers object. Any topos with this property is called a topos with NNO or a WW-topos. The latter term comes from the result that any such topos must have (not only an NNO but also) all W-types.

Toposes over a base

Morphisms of toposes

There are two kinds of homomorphisms between toposes that one considers:

Accordingly there is a 2-category Topos of toposes, whose

Properties

Extensivity

Every topos is an extensive category. For Grothendieck toposes, infinitary extensivity is part of the characterizing Giraud's theorem. For elementary toposes, see toposes are extensive.

Adhesiveness

Every topos is an adhesive category. For Grothendieck toposes this follows immediately from the adhesion of Set, for elementary toposes it is due to (Lack-Sobocinski).

Epimorphisms

In a topos epimorphisms are stable under pullback and hence the (epi, mono) factorization system in a topos is a stable factorization system.

Relation to abelian categories

While crucially different from abelian categories, there is some intimate relation between toposes and abelian categories. For more on that see AT category.

Reasoning in a topos

Any result in ordinary mathematics whose proof is finitist and constructive automatically holds in any topos. If you remove the restriction that the proof be finitist, then the result holds in any topos with a natural numbers object; if you remove the restrictions that the proof be constructive, then the result holds in any boolean topos. On the other hand, if you add the restriction that the proof be predicative in the weaker sense used by constructivists, then the result may fail in some toposes but holds in any Π\Pi-pretopos; if you add the restriction that the proof be predicative in a stronger sense, then the result holds in any Heyting pretopos.

Therefore, one can prove results in toposes and similar categories by reasoning, not about the objects and morphisms in the topos themselves, but instead about sets and functions in the normal language of structural set theory, which is more familiar to most mathematicians. One merely has to be careful about what axioms one uses to get results of sufficient generality.

The internal language of a topos is called Mitchell-Bénabou language.

Examples

Specific examples and key results

Classes of examples

For various applications one uses toposes that have extra structure or properties.

flavors of higher toposes

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\phantom{A}locally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

References

Introductions

Introductions to topos theory include

An introduction amplifying the simple but important case of presheaf toposes is

A quick introduction of the basic facts of Grothendieck topos theory is Chapter I, “Background in topos theory” in

Course notes covering both Grothendieck toposes and the effective topos:

A gentle basic introduction is

Videos

Monographs

Classifying toposes

Specifically on classifying toposes:

Categorical logic and elementary toposes

On categorical logic via toposes:

Algebra, ringed toposes, algebraic geometry

Quantum theory

Original articles

Original source of (Grothendieck) topoi:

That every topos is an adhesive category is discussed in

History

According to appendix C.1 in

“Topos” is a Greek term intended to describe the objects studied by “analysis situs,” the Latin term previously established by Poincaré to signify the science of place [or situation]; in keeping with those ideas, a 𝒰\mathcal{U}-topos was shown to have presentations in various “sites”.

A historical analysis of Grothendieck’s 1973 Buffalo lecture series on toposes and their precedents is in