Topos
Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology.
The mathematical field that studies topoi is called topos theory.
The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic.
Another illustration of the capability of Grothendieck topoi to incarnate the “essence” of different mathematical situations is given by their use as "bridges" for connecting theories which, albeit written in possibly very different languages, share a common mathematical content.
Giraud's theorem already gives "sheaves on sites" as a complete list of examples.
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.
More exotic examples, and the raison d'être of topos theory, come from algebraic geometry.
(of objects given by open subsets and morphisms given by inclusions) whose category of presheaves forms the Zariski topos
But once distinguished classes of morphisms are considered, there are multiple generalizations of this which leads to non-trivial mathematics.
Moreover, topoi give the foundations for studying schemes purely as functors on the category of algebras.
In the case of the étale topos, these form the foundational objects of study in anabelian geometry, which studies objects in algebraic geometry that are determined entirely by the structure of their étale fundamental group.
is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi for the sites
For example, if X is the classifying topos S[T] for a geometric theory T, then the universal property says that its points are the models of T (in any stage of definition Y).
Michael Artin and Barry Mazur associated to the site underlying a topos a pro-simplicial set (up to homotopy).
[4] (It's better to consider it in Ho(pro-SS); see Edwards) 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.
[5] In good cases (if the scheme is Noetherian and geometrically unibranch), this pro-simplicial set is pro-finite.
More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework.
A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid.
Building from category theory, there are multiple equivalent definitions of a topos.
[6] 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.
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.
In the concrete case, namely C(1,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions.
To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics to X as had previously been defined explicitly by the second-order notion of subobject for any category.
For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos.
The Yoneda lemma asserts that Cop embeds in SetC as a full subcategory.
That is, a morphism of graphs can be understood as a pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of generalized elements.
This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of topoi by allowing an object to have multiple underlying sets, that is, to be multisorted.
The category of pointed sets with point-preserving functions is not a topos, since it doesn't have power objects: if
Grothendieck foundational work on topoi: The following monographs include an introduction to some or all of topos theory, but do not cater primarily to beginning students.
