Definition: Let be a category. A Grothendieck topology on is given by associating to each a collection of of -sieves such that

(i) The maximal sieve is in .

(ii) If and is a morphism of , then is in .

(iii) If and is a -sieve such that for any in , then .

A category with a Grothendieck topology is called a site.

Just as in point set topology, we can define a basis to generate a topology on a given space, a Grothendieck pretopology induces a Grothendieck topology, and it is usualy easier not only to specify the former rather than the sieves it generates, but also to verify the sheaf axiom via pretopologies when the category in questions has pullbacks. I will formally state these facts, although I haven’t really gotten into the details of sheaves on sites (but will do shortly).

Proposition 1: Let be a category with pullbacks. Every Grothendieck pretopology on induces a Grothendieck topology on by setting

In this case, we say that is a basis for .

Proposition 2: Let be a category with pullbacks and a Grothendieck topology on with basis . A presheaf is a -sheaf if, and only if, it is a -sheaf.

Proofs can be found in SGL, although the definition of basis is slightly different, and in my dissertation…

Given two Grothendieck topologies on a category we say that is finer than if for every .

Thus the Grothendieck topologies on are partially ordered. The topology in which every sieve is a covering is called the maximal or discrete topology, and the one where only the maximal sieves are coverings is called the minimal or trivial topology, and on this topology every presheaf is a sheaf.

Given a family of Grothendieck topologies on it is clear that is also a Grothendieck topology on , where , and so is the greatest lower bound for . To obtain the lowest upper bound just take the intersection of all the upper bounds for .

In a locally small category, there exists the largest topology for which all the representable functors are sheaves, called the canonical topology. A topology is said to be subcanonical if it is smaller (coarser, or less fine) than the canonical topology.

Second, I said that axiom “(ii) also tells us how to define a sheaf on a category”. Well, it doesn’t. Looking at the sheaf axiom (that temporarily is only in the ‘equalizer’ form) you can see that no cover is actually being “pulled-back”. A sheaf satisfies the sheaf axiom for each covering family independently. The purpose of axiom (ii) is to allow us to use pretopologies as bases for topologies (that shall be defined later).

Third, I defined (as I think SGL also does…) a pretopology in a small category. No such smallness is needed.

Regarding Eduardo’s comment, let’s “put things on clean dishes”, as we say in Brazil.

By set I mean an object of some formal Set Theory, either ZFC, NBG or whatever one uses to define the category Set.

By collection I mean just a plurality of things, which is the notion of set outside any formal theory. Families are usually indexed collections. So a family inside a small category is a set.

And by class I will mean the interpretation of the notion of class inside some formal theory of classes, like NBG, or some construction inside ZFC (using Grothendieck Universes or whatever…).

I do confess to some leniency using such terms in the previous posts…

http://plato.stanford.edu/entries/category-theory/

http://en.wikipedia.org/wiki/Category_theory 

but I realised that they weren’t very appropriate for the non mathematically initiated. Too many examples with groups and abstract topological spaces… And so I thought: I’ve read TONS of introductory material on this matter, maybe it’s time I write my own.

A Brief History of Cats

Mathematicians usually work with one type of structure or class of objects, and most mathematical areas tend to become excessively technical, narrow, closed in on themselves and boring. One of the first mathematicians whose work opposed this was Évariste Galois who, in the 19th century and still in his teens, figured out that to find a formula that gives you the roots of a polynomial equation (like Bhaskara’s does for the quadratic) you must study the possible symmetries of such roots. In order to do that, he defined groups: mathematical structures that generalize the notion of symmetry. So he moved from polynomials to groups – from one type of mathematical object to another. This modus operandi is still active today. In order to understand geometric objects we associate them with algebraic objects and vice-versa. One can count how many “holes” a geometric object has (a sphere has zero, a doughnut, one), or one can take a polynomial and analyse the geometric object that corresponds to its set of zeros (in one dimension they will usually be just a finite number of points, but with more variables we can draw circles and lines). This motivation has lead  to the development of areas know as Algebraic Topology, which associates rigid groups to the ever flexible topological spaces; and Algebraic Geometry, which goes the other way round, associating geometric objects to sets of polynomials.

Category Theory originally arose out of the necessity to better understand such associations between structures so different in nature occurring in Algebraic Topology. In 1945 Samuel Eilenberg and Saunders Mac Lane published “General Theory of Natural Equivalences”, the first paper in which Categories where fully defined. Later on, Eilenberg published (along with Norman Steenrod) a book called Foundations of Algebraic Topology, where categories where used to axiomatize homology theories. 

A decade later Alexander Grothendieck, in my opinion the greatest mathematician of the last half of the 20th century, turned his attention (due to Jean-Paul Serre’s influence) away from Functional Analysis and towards Algebraic Geometry. There, he realized that the most important problems could only be solved if they where correctly stated in the appropriate generality, and thus he appealed to the power of the categorical language, rewriting almost all Algebraic Geometry in category-theoretical terms and basically revolutionising the field.

In the 60′s William Lawvere was trying to put continuum mechanics in a solid mathematical basis, and inspired by the aforementioned cases, also turned to Category Theory. But in doing so his project became more ambitious, and he set out to put all mathematics in a categorical basis using, in a first and not fully satisfactory attempt, the category of all categories. I find quite remarkable that it was in Grothendieck’s work that he found the final solution to this problem. 

Grothendieck, trying to generalize the notion of space, defined the categories known as toposes (plural for topos). These categories have such a rich structure that we can do almost all mathematics in them, and even define formal languages, with their own notions of truth and falsehood. A notion of topos is thus the axiomatization of a mathematical universe. The usual mathematics is done is the universe of sets. But if we regard infinity as philosophically problematic we can do mathematics in a finitary universe. Some have also argued that the law of excluded middle (either something is true or something is false, no in between) is also philosophicaly problematic, and so one does mathematics in an intuitionistic universe. All of these are toposes. And just as after the discovery of non-euclidean geometries we no longer talk about “the” geometry, in a topos-theoretical approach we no longer talk about “the” mathematical universe.

Category Theory has also been used in Physics and occupies a central role in Theoretical Computer Science.

But What Is a Category?

Well, enough with the historical approach, let’s hand-waveingly define things. Category Theory axiomatizes the notion of function and of composition of functions, so we say that a category consists of a bunch o things called arrows, as in

these arrows have a source (A) and a target (B), and if there’s another arrow 

we can align them 

and create another arrow

This is what is called a composition. (At this point, you are probably wondering why the hell and not . Well, worst than stupidity, is canonical stupidity.) This operation is subjected to the following rule, called associativity:

in other words, the order we do things doesn’t matter.

Besides being able to compose arrows, we also have arrows that do nothing, which we call identities. Every arrow has two identities as in 

and composing with them nothing happens:

That’s basically it.

Now, any mathematician can (easily) see that every major area of mathematics is a category. In Set Theory the arrows are functions, in Topology they are continuous functions, in Group Theory homomorphisms, in Linear Algebra they are linear transformations, in Differentiable Geometry they’re smooth maps, and so on… But what’s important is that we shifted from focusing on the objects to focusing on the functions, the ways in which we transform the objects. This is basically the category-theoretical perspective: it’s the functions that matter. 

I’ll leave at this, since I’m afraid I can’t go into any more details without asking too much of the mathematicaly-uneducated reader. To finish of, a rather prophetic quote of Jean Dieudonné:

“… [M]athematics is about to go through a second revolution at this very moment. This is the one which is in a way completing the work of the first revolution, namely, which is releasing mathematics from the far too narrow conditions by ‘set’; it is the theory of categories and functors, for which estimation of its range or perception of its consequences is still too early…”

(i) covers . (identity)

(ii) If covers , and , then covers . (stability)

(iii) If covers , and each is covered by , then covers . (transitivity or local character)

Since is fixed, we can replace the sets by the arrows , and intersections by pullbacks

So instead of families of objects we have families of morphisms called covering families, and we can now restate the previous axioms using arrows, leading to:

Definition: Let be a small category with pullbacks. A Grothendieck pretopology is a map where, for each , is a set of covering families , called -covering families, such that

(i) .

(ii) If and is a morphism of , then .

(iii) If and, for each , , then .

The first thing one may notice is that is actually a contravariant functor, and (ii) tells us how it acts on morphisms. Such functors are called coverages. The second thing one may notice is that (ii) also tells us how to define a sheaf on a category with a Grothendieck Pretopology : A functor is a sheaf if, for every -cover the following diagram is an equalizer

Indeed, coverages are all we need to define a topos of sheaves, but, although essencialy unecessary, the axioms (i) and (iii) do not restrict the theory, for we can prove that for any coverage on a category there is a Grothendieck pretopology that gives us the same sheaves, and the closure properties of (i) and (iii) are useful in practice and most texts usualy include them. For more details and a full exposition of coverages, see Johnstone’s Sketches of an Elephant.

The orinal idea in SGA 4 was that sieve in a category would be a subset such that, if and was a morphism of , then . Sieves where the downwards-connected (if you think the arrows go up…) components of the graph of the category at issue. A -sieve would then be a sieve in the slice category . Since we will only be working with -sieves, we shall proceed directly to their definition and let the reader prove the equivalence to the definiton just mentioned.

Definition: Let be a category and .  A -sieve (or just a sieve) is a family of morphisms of such that,

whenever is defined.

So a sieve is “kinda like” a right ideal…

A -sieve in   is a family of sets such that, if then every open subset of (every open set that is “smaller” that ) is also in .

Now, a rather unexpected fact is that -sieves defined in locally small categories are in one-to-one correspondence with subfunctors of the representable contravariant functor . Given a subfunctor we can define

.

We can readily see that this is indeed a sieve since, given , if , then , for induces a map sending to . Conversly, given a  sieve , we can define a subfunctor by setting .

Given a -sieve , a morphism defines a  -sieve by

.

In terms of functors, we are just taking the pullback of along , i.e. , as in the pullback diagram

where vertical arrow on the right is the natural transformation given by:

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