We note that the notion of an open covering was necessary in order to define the concept of a sheaf, since this was what allowed us to “patch together” the sections of the presheaf over the open subsets of a topological space. So before we can generalize sheaves we must first generalize open coverings and other concepts associated to it, such as intersections.
A product, which is a diagram of objects , , , and morphisms and , and if there is another object and morphisms and , then there is a unique morphism from to such that and . The object is often also referred to as the product and written .
A related notion is that of a pullback (also called a fiber product) is a diagram of objects , , , and , and morphisms , , , and , such that , and if there is another object and morphisms and with , then there is a unique morphism from to such that and . The object is often also referred to as the fibered product and written .
Another related concept is that of a terminal object. A terminal object in a category is just an object such that for every other object in there is a unique morphism .
Finally, we give the definition of an equalizer. We will need this notion when we construct sheaves on our generalization of the open covering of a topological space. An equalizer is a diagram of objects , , and morphisms , , and , such that and if there is another object and morphism such that , there is a unique morphism such that .
By simply reversing the directions of the morphisms on these definitions, we obtain the “dual” notions of coproduct, pushout (also called fiber coproduct), initial object, and coequalizer.
The objects that we have defined above are called universal constructions and are subsumed under the more general concepts of limits and colimits. These universal constructions are unique up to unique isomorphism (An isomorphism in a category is a morphism in for which there exists a necessarily unique morphism in , called the inverse of , such that and ).
These universal constructions are generalizations of familiar concepts. For example, the product in the category of sets corresponds to the cartesian product, while its dual, the coproduct, corresponds to the disjoint union. The terminal object in the category of sets is any set composed of a single element, since every other set has only one function to it, while its dual, the initial object, is the empty set, which has only one function to every other set.
We now proceed with our generalization of an open covering. Let be a category and an object of . A sieve on is given by a family of morphisms on , all with codomain , such that whenever a morphism is in , it is guaranteed that the composition is also in for all morphisms for which the composition makes sense.
If is a sieve on and is any morphism with codomain , then we denote by the family of morphisms with codomain such that the composition is in . is a sieve on .
We now quote from the book Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk the axioms for a Grothendieck topology.
A (Grothendieck) topology on a category is a function which assigns to each object of a collection of sieves on , in such a way that
(i) the maximal sieve is in ;
(ii) (stability axiom) if , then for any arrow ;
(iii) (transitivity axiom) if and is any sieve on such that for all in , then .
The intuitions behind these axioms might perhaps best be seen by considering a category whose objects are open sets and whose morphisms are inclusions of these open sets. Axiom (i) essentially says that the open set is covered by the collection of all its open subsets. Axiom (ii) says that the open subset of is covered by the intersections of with the open subsets covering . Axiom (iii) says that if a collection of open subsets covers every open subset covering , then the collection covers .
We then quote from the same book the definition of a site:
A site will mean a pair consisting of a small category and a Grothendieck topology on . If , one says that is a covering sieve, or that covers (or if necessary, that -covers ).
(The book uses the terminology of a small category to specify that the objects and morphisms of the category form a set, instead of a proper class. The terminology of sets and classes was developed to prevent what is known as “Russell’s paradox” and its variants. In many of the posts on this blog we will not need to explicitly specify whether a category is a small category or not.)
We already know how to construct a presheaf on ; a presheaf is just a contravariant functor from to . Now we just need to generalize the conditions for a presheaf to become a sheaf.
We go back to the conditions that make a (classical) presheaf a sheaf. They can be summarized in the language of category theory by saying that
is the equalizer of
where for and for a family ,
The analogous condition for a (generalized) presheaf on a category equipped with a Grothendieck topology is for
to be an equalizer for
We now introduce the notion of equivalent categories. We first establish some more notation. The set of morphisms from an object to in a category will be denoted by . A functor is called full (respectively faithful) if for any two objects and of , the operation
is surjective (respectively injective). A functor is called an equivalence of categories if it is full and faithful and if any object in is isomorphic to an object in the image of in .
We again refer to the book of Mac Lane and Moerdijk for the definition of a Grothendieck topos:
A Grothendieck topos is a category which is equivalent to the category of sheaves on some site .
A Grothendieck topos is often referred to in the literature as some sort of a “generalized space”. In everyday life we think of “space” as something that objects occupy. Or perhaps we may think of a “place” as something that we live in (the word “topos” itself is the Greek word for “place”). The concept of sheaves expresses the idea that when we look at the objects on portions of a space, they can be “patched together” (it seems rather surreal, even unthinkable, for objects in everyday life not to patch together properly).
We have expressed the notion of a topology as being some sort of “arrangement” on a set. A Grothendieck topology is also an arrangement, but instead of making use of the “parts” (subsets) of a set, it instead makes use of the “relations” or “interactions” between objects in a category.
So we can think of the idea of a topos, perhaps, as making a “place” for our objects of interest (such as sets, groups, rings, modules, etc.) to “live in”. This place has an “arrangement” that our objects of interest “respect”, analogous to how open coverings are used to express how objects are “patched together” to form a sheaf on a topological space. This point of view has already become fruitful in algebraic geometry, where the geometry is described in terms of the algebra; so for instance, the “points” of a “shape” correspond to the prime ideals of a ring (see Rings, Fields, and Ideals and More on Ideals), so they may not correspond with the idea of a space we are usually used to, where the points are described by coordinates which are real numbers.
The idea of making a “place” for mathematical objects to “live in” is abstract enough, however, to not be confined to any one branch of mathematics. Thus, the idea of a topos, sufficiently generalized, has found many applications in everything from logic to differential geometry.
Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk