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 $Sh(X) = Et(X)$ over a topological space $X$: these are the categories of étale spaces over $X$, topological spaces $Y$ that are equipped with a local homeomorphisms $Y \to X$.
When $X = *$ 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 toposes 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 $Set$ 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 $Sh(X)$ as a representative of $X$ itself, while toposes not of this form are “honestly generalized” spaces. This point of view is supported by the fact that the assignment $X\mapsto Sh(X)$ is a full embedding of (sufficiently nice) topological spaces into toposes, and that many topological properties of a space $X$ can be detected at the level of $Sh(X)$. (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 $E$ instead as “toposes in the $E$-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.
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
A quick formal definition is that an elementary topos is a category which
has finite limits,
is cartesian closed, and
has a subobject classifier.
There are alternative ways to state the definition; for instance,
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.
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.
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 $W$-topos. The latter term comes from the result that any such topos must have (not only an NNO but also) all W-types.
There are two kinds of homomorphisms between toposes that one considers:
geometric morphism – this is the kind of morphism that regards a topos as a generalized topological space.
logical morphism – this is the kind of morphism that regards a topos in terms of its internal logic.
Accordingly there is a 2-category Topos of toposes, whose
objects are toposes;
morphisms are geometric morphisms;
2-morphisms are natural transformations between the functors underlying the geometric morphisms.
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).
In a topos epimorphisms are stable under pullback and hence the (epi, mono) factorization system in a topos is a stable factorization system.
While crucially different from abelian categories, there is some intimate relation between toposes and abelian categories. For more on that see AT category.
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 $\Pi$-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.
The archetypical topos is Set. Notice that this happens to be a Grothendieck topos: it is the category of sheaves on the point.
The full subcategory FinSet is also an elementary topos, and the inclusion functor $FinSet \hookrightarrow Set$ is a logical morphism. This is not a Grothendieck topos.
More generally, for $\kappa$ a strong limit cardinal the full subcategory $Set_\kappa$ of sets or cardinality less than $\kappa$ is a topos.
For $C$ any (small) site, the category of sheaves $Sh(C)$ is a Grothendieck topos. Either by definition or by Giraud's theorem, every Grothendieck topos arises in this way. Important examples include:
The case where the Grothendieck topology is the trivial one, so that also all categories of presheaves (on small categories) are (Grothendieck) toposes.
The case of sheaves on (the site given by the category of open subsets of) a topological space, or more generally a locale.
The category $G Set$ of sets equipped with the action of a group $G$: this is the topos of presheaves on the category $\mathbf{B}G$ which is the delooping groupoid of $G$.
For $E$ a topos and $X \in E$ any object, also the overcategory or slice category $E/X$ is again a topos. (Elephant, A.2.3.2). See also over-topos.
If $E$ is a topos and $j \colon E \to E$ is a lex idempotent monad, the category of $j$-algebras is a topos. Each such $j$ corresponds to a Lawvere-Tierney topology in $E$, and the category of $j$-algebras is equivalent to the category of sheaves for this topology. An example is the double-negation topology.
If $G$ is a topological group, then the category $Cont(G)$ of sets with a continuous action of $G$ (that is, the action map $G\times X\to X$ is continuous, where $X$ has the discrete topology) is a topos, and in fact a Grothendieck topos (though this may not be obvious). More generally, $G$ may be a topological groupoid, or even a localic groupoid. In fact, it is a theorem that every Grothendieck topos can be presented as the topos of “continuous sheaves” on a localic groupoid.
Again if $G$ is a topological group, the category $Unif(G)$ of uniformly continuous $G$-sets is also a topos, but not (in general) one of Grothendieck’s. For example, if $G$ is the profinite completion of $\mathbb{Z}$, then a continuous $G$-set is a $\mathbb{Z}$-set all of whose orbits are finite, while a uniformly continuous one is a $\mathbb{Z}$-set with a finite upper bound on the size of all its orbits.
An obvious example of an elementary topos that is not a Grothendieck topos is the category FinSet of finite sets. More generally, for any strong limit cardinal? $\kappa$, the category of sets of cardinality $\lt\kappa$ is an elementary topos, and as long as $\kappa \gt\omega$ it has a NNO. $Set_{\lt \kappa}$ doesn’t even admit a geometric morphism to $Set$.
Since the definition of elementary topos is “algebraic,” there exist free topos?es generated by various kinds of data. In particular, the category of toposes (and logical functors) has an initial object which is sometimes called the free topos. More generally, any higher-order type theory (of the sort which can be interpreted in the internal logic of a topos) generates a free topos containing a model of that theory. Such toposes (for a consistent theory) are never Grothendieck’s.
If $G$ is a large groupoid with a small set of objects, then the category $[G,Set]$ (which makes sense if we define “large” and “small” relative to a Grothendieck universe) is a topos, but not in general a Grothendieck topos. Note that this topos is in fact complete and cocomplete, but it does not have a small generating set, and so is an unbounded topos.
If $\mathcal{F}$ is a filter of subterminal objects in any topos $E$, then there is a filterquotient? topos $E//\mathcal{F}$. Grothendieck-ness (and completeness and cocompleteness) are not in general preserved by this construction.
If $C$ and $D$ are toposes and $F\colon C\to D$ is a lex functor, then there is a topos $Gl(F)$ called the Artin gluing of $C$ and $D$ along $F$, and defined to be the comma category $(D/F)$. If $C$ and $D$ are Grothendieck toposes then $Gl(F)$ is a $Set$-topos. If $F$ is accessible, then $Gl(F)$ is again Grothendieck (hence bounded), but in general it may not be. (Note, though, that it is not clear whether it can be proven in ZFC that there exist any inaccessible lex functors between Grothendieck toposes, although from a proper class of measurable cardinals one can construct an inaccessible lex endofunctor of $Set$.)
The category of coalgebras for any lex comonad on a topos is again a topos: a topos of coalgebras, and if the comonad is accessible, this construction preserves Grothendieck-ness. If the comonad is not accessible, then this topos is unbounded.
For instance the Artin gluing $Gl(F)$ is equivalent to the category of coalgebras for the comonad on the topos $C\times D$ defined by $(c,d) \mapsto (c, d\times F(c))$.
More generally, the category of coalgebras for any pullback-preserving comonad on a topos is again a topos. This covers both the preceding result and also the overcategory (slice category) result above, since $E/X$ is the category of coalgebras for the pullback-preserving comonad given by $X \times - \colon E \to E$. It also covers Artin gluing along a pullback-preserving functor.
More generally still, Todd Trimble has a notion called a “modal operator” on a topos, from which one can construct a new topos of ”$G$-structures”: see Three topos theorems in one. Special cases include the pullback-preserving comonad result just mentioned, and the result that the category of algebras for a lex idempotent monad is a topos. A related idea is Toby Kenney’s notion of lex distributive diad?, from which one can also construct a topos.
From any partial combinatory algebra one can construct a realizability topos, whose internal logic is “computable” or “effective” mathematics relative to that PCA. In particular, for Kleene's first algebra, one obtains the effective topos, the most-studied of realizability toposes. Realizability toposes are generally not Grothendieck toposes.
A topos can also be constructed from any tripos. This includes realizability toposes as well as toposes of sheaves on locales.
For various applications one uses toposes that have extra structure or properties.
In the foundations of mathematics, one often studies well-pointed toposes, especially models of ETCS as potential replacements for the category Set.
In synthetic differential geometry one studies smooth toposes as a context for axiomatic differential geometry.
Locally presentable categories: Large categories whose objects arise from small generators under small relations.
(n,r)-categories | satisfying Giraud's axioms | inclusion of left exaxt localizations | generated under colimits from small objects | localization of free cocompletion | generated under filtered colimits from small objects | ||
---|---|---|---|---|---|---|---|
(0,1)-category theory | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | ||
category theory | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories | $\hookrightarrow$ | accessible categories |
model category theory | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | ||
(∞,1)-topos theory | (∞,1)-toposes | $\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories | $\hookrightarrow$ | accessible (∞,1)-categories |
Introductions to topos theory include
Ross Street, A survey of topos theory (notes for students, 1978) pdf
Tom Leinster, An informal introduction to topos theory (2010)
An introduction amplifying the simple but important case of presheaf toposes is
A standard textbook is
This later grew into the more detailed
A quick introduction of the basic facts of Grothendieck topos theory is chapter I, “Background in topos theory” in
A standard textbook on this case is
There is also
A gentle basic introduction is
A quick introduction of the basic facts of Grothendieck topos theory is chapter I, “Background in topos theory” in
A survey is in
That every topos is an adhesive category is discussed in