Cats in the Jungle

Grothendiek Topologies – Part II

•November 23, 2008 • 1 Comment

Now, although pretopologies generalize the notion of open coverings, we cannot recover the pretopology by specifying exactly which presheaves are sheaves, for the notion of a pretopology being “finer” than another is as well behaved as in the set theoretical case. For instance, if a presheaf $\mathcal{P}$ satisfies the sheaf axiom for a covering family $K$ , it will also satisfy it for any covering family $L$ that contains $K$ , and the converse is true if every morphism of $L$ factors through one of $K$ . In order to avoid this we turn to coverings that are downward saturated, in other words, sieves.

Definition: Let $\mathcal{C}$ be a category. A Grothendieck topology on $\mathcal{C}$ is given by associating to each $U \in \mathcal{C}$ a collection of $J(U)$ of $U$ -sieves such that

(i) The maximal sieve $M_U$ is in $J(U)$ .

(ii) If $S \in J(U)$ and $f: V \to U$ is a morphism of $\mathcal{C}$ , then $f^*S$ is in $J(V)$ .

(iii) If $S \in J(U)$ and $Q$ is a $U$ -sieve such that $f^*Q \in J(V)$ for any $f:V \to U$ in $S$ , then $Q \in J(U)$ .

A category with a Grothendieck topology $(\mathcal{C},J)$ 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 $\mathcal{C}$ be a category with pullbacks. Every Grothendieck pretopology $K$ on $\mathcal{C}$ induces a Grothendieck topology $J$ on $\mathcal{C}$ by setting

$S \in J(U) \Leftrightarrow \exists R \in K(U); R \subset S.$

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

Proposition 2: Let $\mathcal{C}$ be a category with pullbacks and $J$ a Grothendieck topology on $\mathcal{C}$ with basis $K$ . A presheaf $\mathcal{P} : \mathcal{C}^{op} \to \textbf{Set}$ is a $J$ -sheaf if, and only if, it is a $K$ -sheaf.

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

Given two Grothendieck topologies $J, J'$ on a category $\mathcal{C}$ we say that $J$ is finer than $J'$ if $J'(U) \subseteq J(U)$ for every $U \in \mathcal{C}$ .

Thus the Grothendieck topologies on $\mathcal{C}$ 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 $\{J_\alpha\}$ of Grothendieck topologies on $\mathcal{C}$ it is clear that $\bigcap J_\alpha$ is also a Grothendieck topology on $\mathcal{C}$ , where $\left( \bigcap J_\alpha \right)(U) = \bigcap J_\alpha(U)$ , and so $\bigcap J_\alpha$ is the greatest lower bound for $\{J_\alpha\}$ . To obtain the lowest upper bound just take the intersection of all the upper bounds for $\{J_\alpha\}$ .

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.

Posted in Sheaves

Errata: Grothendieck Topologies – Part I

•November 17, 2008 • Leave a Comment

First, I led to believe that a coverage was defined as functor. Well, it isn’t. That’s just a property it has in small categories. It is defined as a function that associates objects with families of morphisms and satisfies axiom (ii) of pretopologies.

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…

Posted in Sheaves

Category Theory for Non-Mathematicians

•November 15, 2008 • Leave a Comment

I’ve written this post mainly because a friend of mine asked me what my master’s dissertation was about, and anyone who knows me well can predict that I didn’t just say “Oh, it’s this really complicated thing called Topos Theory. You’ll never understand it.” and left it at that. I started one of my usual inspired and passionate monologues, and, since her own area of interest (lacanian psychoanalysis, I presume…) has lead her into excursions through mathematics and its philosophy, she became curious and asked me for some references. I was going to suggest these,

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

$A \stackrel{f}{\longrightarrow} B$

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

$B \stackrel{g}{\longrightarrow} C$

we can align them

$A \stackrel{f}{\longrightarrow} B$ $\stackrel{g}{\longrightarrow} C$

and create another arrow

$A \stackrel{g \circ f}{\longrightarrow} C$

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

$(h \circ g) \circ f =$ $h \circ ( g \circ f )$

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

$A \stackrel{I_A}{\longrightarrow}$ $A \stackrel{f}{\longrightarrow} B$ $\stackrel{I_B}{\longrightarrow} B$

and composing with them nothing happens:

$f \circ I_A = f = I_B \circ f$

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…”

Posted in Uncategorized

Grothendieck Topologies – Part I (Pretopologies)

•November 10, 2008 • 3 Comments

Given an open set $U \in X$ , the three properties of open coverings of topological spaces that shall be generalized are:

(i) $\{U\}$ covers $U$ . (identity)

(ii) If $\{U_\alpha\}$ covers $U$ , and $V \subseteq U$ , then $\{V \cap U_\alpha\}$ covers $V \cap U$ . (stability)

(iii) If $\{U_\alpha\}$ covers $U$ , and each $U_\alpha$ is covered by $\{V_{\alpha \beta}\}_{\beta \in B}$ , then $\bigcup_\beta \{V_{\alpha\beta}\}$ covers $U$ . (transitivity or local character)

Since $U$ is fixed, we can replace the sets $V, U_\alpha \subseteq U$ by the arrows $V, U_\alpha \to U$ , and intersections $V \cap U_\alpha$ by pullbacks

$\begin{array}{ccc} V \times_U U_\alpha & \to & U_\alpha\\ \downarrow & & \downarrow \\ V & \to & U \end{array}$

So instead of families of objects $\{U_\alpha\}$ we have families of morphisms $\{U_\alpha \to U\}$ called covering families, and we can now restate the previous axioms using arrows, leading to:

Definition: Let $\textbf{C}$ be a small category with pullbacks. A Grothendieck pretopology is a map $P: \textbf{C} \to \textbf{Set}$ where, for each $U \in \textbf{C}$ , $P(U)$ is a set of covering families $\{ U_\alpha \to U \}$ , called $P$ -covering families, such that

(i) $\{ I_U: U \to U \} \in P(U)$ .

(ii) If $\{U_\alpha \to U \} \in P(U)$ and $f: V \to U$ is a morphism of $\textbf{C}$ , then $\{ \pi_1: V \times_U U_\alpha \to V \} \in P(V)$ .

(iii) If $\{U_\alpha \to U \} \in P(U)$ and, for each $U_\alpha$ , $\{ V_{\alpha\beta} \to U_\alpha\} \in P(U_\alpha)$ , then $\{ V_{\alpha \beta} \to U_\alpha \to U\} \in P(U)$ .

The first thing one may notice is that $P: \textbf{C} \to \textbf{Set}$ 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 $(\textbf{C},P)$ : A functor $F : \textbf{C}^{op} \to \textbf{Set}$ is a sheaf if, for every $P$ -cover $\{U_\alpha \to U\}$ the following diagram is an equalizer

$F(U) \to \displaystyle{\prod_{\alpha \in A}} F(U_\alpha) \rightrightarrows \prod_{\alpha ,\beta \in A} F(U_\alpha \times_U U_\beta)$

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.

Posted in Sheaves

Sieves

•November 8, 2008 • 2 Comments

In order to define sheaves over arbitrary categories, not just the usual $\mathcal{O}(X)^{op}$ , we will need some way of generalizing the notion of a topology on a given space. Grothendieck’s insight was that what really mattered where not the open sets themselves but open coverings. But only specifying families of open coverings for the elements of a category can lead to ambiguities, since two different sets of covering families can give rise to the same category of sheaves (just like we can get the same topology from two different basis for that topology). In order to remove this problem we must work with coverings that are in some way saturated, and this is where sieves come in.

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

Definition: Let $\mathcal{C}$ be a category and $U \in \mathcal{C}$ . A $U$ -sieve (or just a sieve) is a family of morphisms $S_U$ of $\mathcal{C}$ such that,

$f \in S_U \Rightarrow f \circ g \in S_U,$

whenever $f\circ g$ is defined.

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

A $U$ -sieve in $\mathcal{O}(X)$ is a family of sets $S_U = \{ V \in \mathcal{O}(X) : V \subseteq U \}$ such that, if $V \in S_U$ then every open subset of $V$ (every open set that is “smaller” that $V$ ) is also in $S_U$ .

Now, a rather unexpected fact is that $U$ -sieves defined in locally small categories are in one-to-one correspondence with subfunctors of the representable contravariant functor $Hom( -, U): \mathcal{C} \to \textbf{Set}$ . Given a subfunctor $F \subseteq Hom(-,U)$ we can define

$S_U = \{ f \in Mor(\mathcal{C}) : \exists V; f \in F(V) \}$ .

We can readily see that this is indeed a sieve since, given $g: W \to V$ , if $f \in F(V)$ , then $f \circ g \in F(W)$ , for $g$ induces a map $F(g) : F(V) \to F(W)$ sending $f$ to $F(g)(f) = g_* (f) = f \circ g$ . Conversly, given a sieve $S_U$ , we can define a subfunctor by setting $F(V) = \{ f \in S_U : dom(f) = V \}$ .