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Thursday, May 19, 2005
Continuing on from my earlier discussion of category theory, I present:
Topos Theory!
The recent major textbook on Topos Theory likens the subject to that story of people trying to describe an elephant to a blind man: Topos Theory is a lot of different things to a lot of different people. It is a generalization of topology, a new way of expressing logic, a foundation for quantum mechanics...but to me, there is one aspect of it that I think is the most important: a topos is a generalization of sets.
If you think about how, philsophically, mathematics has progressed over the past couple hundred years, the major theme is abstraction. One takes a structure, generalizes it's properties, and studies the structures which have those properties. The reason for the generalizations are to attempt to get a greater understanding of the original structure, by removing those properties not considered important to the structure.
Through all these generalizations, one thing has remained constant: the idea that structures are just sets with properties. But if you think about it, why should we live with one single fixed universe of sets? It's a bit like doing linear algebra over a fixed basis, rather than simply considering some abstract basis. Same idea here: why not try and abstract out the properties of sets and functions, then work in that generalized setting?
The tool that we can use to do this is category theory. By considering set, the category of sets and functions, we can try and abstract the properties of set to a more general form of category: a topos. So a topos will be a category with certain properties that allow one to work with it as one does set. The objects of a topos will correspond to sets, and the arrows of the topos will correspond to functions. Then instead of considering, say, sets with an certain properties, we can consider objects of toposes with certain properties. As with any abstraction, by working in this more general setting we can see what the essential facts and properties of sets are. In particular, we can try and see whether things like the axiom of choice are important.
Fortunately, when people started looking at this idea of abstracting set, there were already enough categorical notions to be able to understand the properties of set they needed to abstract. There are three axioms for a topos, with the third being probably the most important.
(1) Existence of Finite Limits. A limit in a category is not really like a limit in analysis. Rather, a limit is a sort of "solution object" to a set of equations. An example of this is the equalizer of two functions. Sps one had two functions f, g: A -> B. The "equalizer" of f and g would be the set of a in A for which f(a) = g(a). So if a category has equalizers, then for any two arrows f, g: A -> B, there is an object E with an arrow h: E -> A such that fh = gh, and E is "universal" with this property. In the case of set, the equalizer E would be, as described above, the set of a in A with the property that f(a) = g(a). The map h: E -> A is simply inclusion.
Saying that a category has has "finite limits" then means that for any kind of finite arrangement of arrows there is some object which is a "solution" for the system of arrows. In particular, this also includes the case of products, as a sort of base case. In other words, in a topos one can form the product of two objects AxB, as described in my first post.
(2) Existence of Exponentials. This is a bit easier to understand. In set, for any two sets A and B, A^B usually denotes the set of functions of B -> A. We would like to have the same thing in a topos, namely that there is some sort of object A^B which corresponds to the set of arrows from B to A.
(3) Existence of a Subjobject Classifier: The Truth Object. This axiom I think is really quite brilliant. The idea is to have in a topos a way of characterizing when something is a subobject of something else. One does this by having a "truth" object. In set, the truth object is the two element set {0, 1}, or {f, t} (f for false, t for true). It is pretty standard how this is used to characterize a subset: a subset A of B is the same thing as a function from B to {0, 1}: the function sends b to 0 if it is not in A, and 1 if it is in A. In other words, subsets are the same thing as characteristic functions. One requires the exact same thing in a topos: a truth object "Omega" which has the property that subobjects of A are the same thing as arrows from A to Omega.
The reason these three axioms are considered enough is that one can show that any sentence one writes down in set theory can then be expressed in a general topos. Another reason these axioms are considered good is that there are an abundance of examples of categories with these three properties. An example is func, the category of functions and commutative squares. In this category, the objects are functions f, and an arrow from a function f to a function g is a pair of functions (h, k) such that kf = gh (if one writes out what this looks like, each arrow (h, k) forms a square with the functions f and g). This category is a topos. What is really fascinating is it's truth object: it is the function {0, 1} -> {0, 1, 2}. No longer is something simply true or false, now it can have a degree of truth or falsity in this topos.
In fact, in most toposes the truth object Omega is a lot more complicated than simply being {false, true}. Even more fascinating is that this means that in most toposes, the law of the excluded middle does not hold! In other words, one can not perform proof by contradiction in a general topos. Moreover, it can be shown that the axiom of choice has meaning in a topos, and that a topos with the axiom of choice must have the law of excluded middle. So most toposes do not have either the law of the excluded middle or the axiom of choice.
To many people, of course, this is akin to heresy; certainly I was shocked to hear this. How could a general theory of "sets" not allow one to prove things by contradiction? On the other hand, the axioms for a topos are so very simple, elegant, and work so well that one has a hard time not accepting them. It will be a major project for many years to come for people to work out how to be able to do math in a general topos, but I think it may be a very worthy project, as any abstraction usually is.
Topos Theory!
The recent major textbook on Topos Theory likens the subject to that story of people trying to describe an elephant to a blind man: Topos Theory is a lot of different things to a lot of different people. It is a generalization of topology, a new way of expressing logic, a foundation for quantum mechanics...but to me, there is one aspect of it that I think is the most important: a topos is a generalization of sets.
If you think about how, philsophically, mathematics has progressed over the past couple hundred years, the major theme is abstraction. One takes a structure, generalizes it's properties, and studies the structures which have those properties. The reason for the generalizations are to attempt to get a greater understanding of the original structure, by removing those properties not considered important to the structure.
Through all these generalizations, one thing has remained constant: the idea that structures are just sets with properties. But if you think about it, why should we live with one single fixed universe of sets? It's a bit like doing linear algebra over a fixed basis, rather than simply considering some abstract basis. Same idea here: why not try and abstract out the properties of sets and functions, then work in that generalized setting?
The tool that we can use to do this is category theory. By considering set, the category of sets and functions, we can try and abstract the properties of set to a more general form of category: a topos. So a topos will be a category with certain properties that allow one to work with it as one does set. The objects of a topos will correspond to sets, and the arrows of the topos will correspond to functions. Then instead of considering, say, sets with an certain properties, we can consider objects of toposes with certain properties. As with any abstraction, by working in this more general setting we can see what the essential facts and properties of sets are. In particular, we can try and see whether things like the axiom of choice are important.
Fortunately, when people started looking at this idea of abstracting set, there were already enough categorical notions to be able to understand the properties of set they needed to abstract. There are three axioms for a topos, with the third being probably the most important.
(1) Existence of Finite Limits. A limit in a category is not really like a limit in analysis. Rather, a limit is a sort of "solution object" to a set of equations. An example of this is the equalizer of two functions. Sps one had two functions f, g: A -> B. The "equalizer" of f and g would be the set of a in A for which f(a) = g(a). So if a category has equalizers, then for any two arrows f, g: A -> B, there is an object E with an arrow h: E -> A such that fh = gh, and E is "universal" with this property. In the case of set, the equalizer E would be, as described above, the set of a in A with the property that f(a) = g(a). The map h: E -> A is simply inclusion.
Saying that a category has has "finite limits" then means that for any kind of finite arrangement of arrows there is some object which is a "solution" for the system of arrows. In particular, this also includes the case of products, as a sort of base case. In other words, in a topos one can form the product of two objects AxB, as described in my first post.
(2) Existence of Exponentials. This is a bit easier to understand. In set, for any two sets A and B, A^B usually denotes the set of functions of B -> A. We would like to have the same thing in a topos, namely that there is some sort of object A^B which corresponds to the set of arrows from B to A.
(3) Existence of a Subjobject Classifier: The Truth Object. This axiom I think is really quite brilliant. The idea is to have in a topos a way of characterizing when something is a subobject of something else. One does this by having a "truth" object. In set, the truth object is the two element set {0, 1}, or {f, t} (f for false, t for true). It is pretty standard how this is used to characterize a subset: a subset A of B is the same thing as a function from B to {0, 1}: the function sends b to 0 if it is not in A, and 1 if it is in A. In other words, subsets are the same thing as characteristic functions. One requires the exact same thing in a topos: a truth object "Omega" which has the property that subobjects of A are the same thing as arrows from A to Omega.
The reason these three axioms are considered enough is that one can show that any sentence one writes down in set theory can then be expressed in a general topos. Another reason these axioms are considered good is that there are an abundance of examples of categories with these three properties. An example is func, the category of functions and commutative squares. In this category, the objects are functions f, and an arrow from a function f to a function g is a pair of functions (h, k) such that kf = gh (if one writes out what this looks like, each arrow (h, k) forms a square with the functions f and g). This category is a topos. What is really fascinating is it's truth object: it is the function {0, 1} -> {0, 1, 2}. No longer is something simply true or false, now it can have a degree of truth or falsity in this topos.
In fact, in most toposes the truth object Omega is a lot more complicated than simply being {false, true}. Even more fascinating is that this means that in most toposes, the law of the excluded middle does not hold! In other words, one can not perform proof by contradiction in a general topos. Moreover, it can be shown that the axiom of choice has meaning in a topos, and that a topos with the axiom of choice must have the law of excluded middle. So most toposes do not have either the law of the excluded middle or the axiom of choice.
To many people, of course, this is akin to heresy; certainly I was shocked to hear this. How could a general theory of "sets" not allow one to prove things by contradiction? On the other hand, the axioms for a topos are so very simple, elegant, and work so well that one has a hard time not accepting them. It will be a major project for many years to come for people to work out how to be able to do math in a general topos, but I think it may be a very worthy project, as any abstraction usually is.
