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| Guitarist |
 Posted: Thu Jun 12, 2008 9:37 am Post subject: Category Theory Primer |
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What follows scarcely seems like mathematics at all. Real mathematicians frequently recoil at the extreme level of abstraction involved in this subject, but I maintain it is one that can be enjoyed by those will very little math training. In a sense, it is on the interface between math and philosophy. (I say this advisedly, as I hope to make clear).
So, You have all put up with my long-winded and boring dissertations on things like Groups, Vector Spaces, Topological Spaces, Manifolds etc. Actually, it comes as some surprise to learn that many of these constructions are relatively modern - one might say that the history of 19th and 20th century has been one of increasingly finer dissection of the subject into sub-disciplines.
Then, in 1945 a guy named Saunders Mac Lane asked the seemingly simple question: Is it possible to come up with a single language that will describe all of of these disciplines with equal accuracy? He decided the answer is "Yes", and that this language should be called category theory.
So we had better start with some definitions, which I motivate as follows. When we do set theory, we handle "objects" called sets, with "arrows" pointing form one object to another called set functions. When we do group theory, we handle objects called groups, with arrows between them, called group homomorphisms, vector spaces have linear transformations as arrows etc etc..
So, my definitions:
A category C consists of a collection of mathematical objects X, Y, Z for which the following is true;
for any pair of objects X, Y, there is at least one "mapping" f: X → Y called a morphism, or, by me, an arrow;
for each object X ∈ C there is a privileged arrow called the identity, idX: X → X;
if f: X → Y, and g: Y → Z, then g ⋅ f: X → Z; this composition is associative;
f ⋅ idX = f and f = idY ⋅ f.
That's it!
Oh - notice that I compose arrows right-to-left, this is standard.
So, I will give a couple of examples, make an important comment, then leave you in peace.
In the category Set, the objects are sets, the arrows are set functions. In the category Grp the objects are groups, the arrows are group homomorphisms. In the category K-Vec the objects are vector spaces over the field K, the arrows are linear transformations. Simple really.
Now note this well; category theory is not really interested in the objects themselves, rather the arrows are the main interest - I will show you why later, with an example.
If objects take back seat, the elements in the objects - set elements, group elements, vectors, in our examples above - don't even make it through the door! |
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| serpicojr |
| Posted: Thu Jun 12, 2008 10:33 am Post subject: |
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Welcome back! This looks like it could be a great thread--it may drum up a lot of interest in the math forum.
One small nitpick: in my definition of a category, there need not exist a morphism between any pair A, B of objects. This isn't that important, because most applications involve objects which always have some sort of trivial map--a 0 map in an algebraic setting, a point map in a topological one, etc.
As an example of a category in which not all pairs of objects have morphisms between them, consider a poset (P,≤)*. Define a category where the objects are the points of P and there is a morphism between a and b iff a≤b. Check that this is a category. If we take a poset (P,≤) in which there are incomparable elements (i.e., elements a and b so that neither a≤b or b≤a is true), then not all pairs of objects have morphisms between them. For example (N,|), the natural numbers under divisibility, satisfies this, as any two distinct primes are incomparable.
*P is a set, ≤ is a reflexive, transitive binary relation on P, and ≤ is antisymmetric--if a≤b and b≤a, then a=b.
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PS: I guess you can always turn any category into one which always has morphisms between any two objects. Let's take a category C and, between any two objects A, B of C, throw in a morphism Z: A -> B. Declare that Z composed on either side with any other morphism gives the Z morphism between the appropriate objects. All of your old info is still there.
Last edited by serpicojr on Thu Jun 12, 2008 10:41 am; edited 1 time in total |
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| accountabled |
| Posted: Thu Jun 12, 2008 10:38 am Post subject: |
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Joined: 29 Oct 2007
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Nice!
I really like the approach of taking a macroscopic view rather than a specialised perspective. Taking a step back and looking for similarities and pattern in all fields of modern mathematics can definitely yield new insights. I'm not a trained mathematician, but I actually can follow your logic (I think). By generalizing some of the common themes between set, vector and group theory, one can benefit from a more universal big picture. It can help remove intellectual thresholds and thereby accellerates learning (I personally struggle to understand details of a subject without being able to relate them to some kind of global mental framework that consist of no more than 6-8 building blocks).
From a philosophical perspective the following question immediately bubbled up:
Is there in your model also a Category of all Categories ? |
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| serpicojr |
| Posted: Thu Jun 12, 2008 10:47 am Post subject: |
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| Excellent question! I don't want to step on Guitarist's toes, so let me just say that there is indeed a notion of morphism between two categories, and these morphisms are called functors. |
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| Guitarist |
| Posted: Thu Jun 12, 2008 1:01 pm Post subject: |
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Forum Ph.D.
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| serpicojr wrote: |
| Welcome back! |
Thanks, it's been a while, I know.
| Quote: |
| As an example of a category in which not all pairs of objects have morphisms between them, consider a poset (P,≤) |
Yeah, right, this was to be my canonical example of why category theory cares nothing for elements. I'll get to this directly.
Accountabled: Yes, there is a category of all categories. Unsurprisingly, it's called Cat. In fact that is one of the reasons I called a category a collection of objects. If I called it a set, I would incur the wrath of Russell, precisely for this reason (well, among others)!
More tomorrow, if you can bear it....... |
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| Guitarist |
| Posted: Fri Jun 13, 2008 6:18 am Post subject: |
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OK, kids, let's press on. serpicojr raised an interesting point, which I now want to explain. I warn you, this had me tugging my beard for a full day (though, as you're all probably smarter than me, maybe you'll breeze through it.). Either way, take it slow and ask questions, as it is rather subtle.
often there is more that one way to "categorize" a mathematical object, which I will illustrate for the case case of posets, since our friend raised them.
Recall first the poset ( = partially ordered set) axioms;
x ≤ x (reflexivity)
if x ≤ y, then y ≥ x (antisymmetry)
if x≤ y and y ≤ z, then x ≤ z (transitivity).
Now we need a certain generosity of spirit to interpret the symbol ≤. It may, of course mean the usual arithmetic "less than or equal to". But we will also take it to include the relation ⊆, and also the relation "x divides y", often written as x|y. Bear this in mind, as I shall return to this shortly.
Now, I can consider the category of posets, let's call it Pos, whose object are partially ordered set in the above loose sense, and whose arrows (morphisms, if you prefer) are monotone set functions. (recall that a function is called monotone if, roughly speaking, it preserves order in the above sense of the word.
In this sense, serpico was not quite right.
BUT.... I can also interpret the relations x ≤ y, x ⊆ y, x|y as an arrow x → y. This implies that I may treat a randomly chosen poset as a category unto itself, whose category theoretic objects are set elements, and whose arrows are the poset binary relations.
Now since the poset axioms do not permit a total order (2|14 and 7|14, does 2|7?), serpico was right in this context; not all objects in this category have arrows between.
But note that, I may have as many such posets as I choose, each having different partially ordered set elements. But there is only one such category!!. This is why I say I care eff-all about objects, since in fact, this latter category should perhaps be referred to as the category of ≤'s.
It so happens the same reasoning is true for preorders, monoids and groups.
Maybe more later, but I'd better get back to work...... |
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| serpicojr |
| Posted: Fri Jun 13, 2008 8:55 am Post subject: |
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| Guitarist wrote: |
| But note that, I may have as many such posets as I choose, each having different partially ordered set elements. But there is only one such category!! |
I'm not sure what you mean by this. |
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| Guitarist |
| Posted: Fri Jun 13, 2008 9:51 am Post subject: |
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OK, maybe I expressed myself poorly. Let's try again.
Consider the category Pos of all partially ordered sets, whose morphisms are monotone set functions. The number of such objects in Pos is huge.
By my earlier assertion, that I am free to interpret the relation ≤ as a category theoretic morphism, then it may appear that, for any and all objects in Pos, there is again a category, whose objects are set elements, and whose morphisms are ≤, and that is there is therefore a correspondingly huge number of such categories.
But this is simply not the case - there is only one such category. Therefore, I conclude that I had better not worry too much about the objects, rather accept that by the arrow ≤ = →, I may mean less than or equal, set inclusion or divisible mod zero, or whatever else springs to mind, and that is all that counts.
You may well ask then "what is this category called, if it isn't quite Pos?" Well the answer is disappointing, I'm afraid; it's just called a category, since category theory, at it's deepest level, is simply a generalization of the concepts of preorder and monoid.
I do confess, this concept caused me no end of grief at first |
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| serpicojr |
| Posted: Fri Jun 13, 2008 4:33 pm Post subject: |
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| Guitarist wrote: |
| Consider the category Pos of all partially ordered sets, whose morphisms are monotone set functions. The number of such objects in Pos is huge. |
Yes.
| Quote: |
| By my earlier assertion, that I am free to interpret the relation ≤ as a category theoretic morphism, then it may appear that, for any and all objects in Pos, there is again a category, whose objects are set elements, and whose morphisms are ≤, and that is there is therefore a correspondingly huge number of such categories. |
Yes.
| Quote: |
| But this is simply not the case - there is only one such category. |
No. You just constructed a category for each poset. Once you introduce the notions of equivalence of categories, you can show that each category defined by a poset is inequivalent to any other category defined by a nonisomorphic poset. |
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| JaneBennet |
| Posted: Fri Jun 13, 2008 5:18 pm Post subject: |
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Category theory is great at abstracting a lot that is common between abstract structures that may look different at first glance, but the attempt at looking for similarities shouldnât be carried too far. A particular category may have certain properties that are unique to itself and not easily generalizable to other categories.
Take, for instance, the question of isomorphisms â or âiso-arrowsâ if you like. An iso-arrow from one object X to another object Y is an bijective arrow from X to Y such that the inverse mapping from Y to X is also an arrow. In certain categories, it is sufficient for an arrow to be bijective in order to become an iso-arrow. This is the case for groups and rings, where the arrows are homomorphims and the iso-arrows are isomorphisms: any bijective homomorphism is an isomorphism. Itâs also the case for vector spaces, where the arrows are linear transformations and the iso-arrows are isometries (distance-preserving linear transformations). For topological spaces, things are a bit different. Here the arrows are continuous functions and the iso-arrows are homeomorphisms â but bijective continuous functions are not necessarily homeomorphisms.
This is because a bijection between two topological spaces is not necessarily bicontinuous: it may be continuous in one direction but not in the other. For example, let X be the topological space consisting of the real numbers ℝ with the power set of ℝ and Y be the topological space consisting of ℝ with the topology {Ă, ℝ}. If Φ is the identity map of the real numbers (so it is its own inverse), then Φ:X→Y is continuous but Φ:Y→X is not continuous. Hence Φ is not a homeomorphism. (Indeed X and Y are clearly nonhomeomorphic since Y is a connected space whereas X isnât.)
For groups and rings, if a homomorphism is bijective, the inverse function is automatically a homomorphism as well. Similarly, if a linear transformation between vector spaces is a bijection, its inverse is always a linear transformation. Continuous functions between topological spaces are different. 
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| serpicojr |
| Posted: Fri Jun 13, 2008 6:04 pm Post subject: |
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| JaneBennet wrote: |
| Itâs also the case for vector spaces, where the arrows are linear transformations and the iso-arrows are isometries (distance-preserving linear transformations). |
Careful--not all vector spaces come equipped with a metric, and so the notion of isometry doesn't always make sense in a vector space. And an isometry need not always be invertible. For example, consider the vector space of functions from N to C for which the series with terms f(n) is absolutely convergent. Define a norm on this space to be the absolute value of the sum of the series with terms f(n). If you map f to the function g(n) = f(n-1) for n > 1, g(1) = 0, you have that f and g have the same norm, so this is an isometry. However, g sits in the proper subspace of functions with g(1) = 0.
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PS: And I should have also stated that not all invertible linear maps are isometries--for example, scalar multiplication is a linear isomorphism, but this is only an isometry when the scalar has absolute value 1. (I guess I'm assuming we're working in a normed space here.) |
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| JaneBennet |
| Posted: Sat Jun 14, 2008 4:28 am Post subject: |
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Oops. I was thinking of rotations and reflections in ℝ2 and ℝ2, forgetting that these are also metric spaces. 
Whatâs the name for two vector spaces that are âessentially the sameâ (i.e. for which there exists a bijective linear transformation between them)? Groups/Rings which are essentially the same are isomorphic, and topological spaces which are essentially the same are homeomorphic. What about vector spaces which are essentially the same? 
_________________
A problem worthy of attack
Proves its worth by fighting back.
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Fact of the day: Old English
Last edited by JaneBennet on Sat Jun 14, 2008 5:03 am; edited 2 times in total |
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| Guitarist |
| Posted: Sat Jun 14, 2008 4:48 am Post subject: |
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| serpicojr wrote: |
No. You just constructed a category for each poset. Once you introduce the notions of equivalence of categories, you can show that each category defined by a poset is inequivalent to any other category defined by a nonisomorphic poset. |
Well OK, I guess I was shortcutting somewhat. But it does illustrate an important point;
militant category theorists look askance at words like "equal" and "the same", rather, under a natural isomorphism, two or more objects or categories are sufficiently alike to be considered as one.
My purpose had been to try and illustrate the point that the exact nature of the poset elements, in this case regarded as objects in a category, is entirely immaterial, which brings me to
Jane: The whole point of category theory, to my mind at least, is that, anything I say about one category can usually (not always) be taken to apply any other category.
So words like "bijection", "norm" etc. are not recognized, though there is a related category theoretic notion for injection, surjection and bijection. We can talk some about that later, but foe now lemme just stamp out a few more definitions.
Let C and D be arbitrary categories. The mapping F: C → D is called a functor. (i was about to say this gadget is unique to category theory, but, historically at least, that's not quite correct; when Poincare introduced us to the Fundamental Group, for example, he did so via a mapping of this sort from a pointed toplogolical space to the homotopy group. In modern lingo we might write Top* → Grp).
Anyhoo - functors are very respectful creatures; they respect objects, they respect the identity, they respect arrows, they respect composition of arrows etc.
So, for example, in the case that, say f: X → Y ∈ C, then F(f): F(X) → F(Y) ∈ D, and so on.
Now it might immediately occur to you, as it did to me, that there's a problem here; what about the functor F: Set → Grp, say. How can a functor magically conjure the group axioms simply by being fed a set? Ah well, boys and girls, that will have to wait. I have something more important to do first.
Recall I was at pains not to refer to the collection of objects that comprise a category as a set. Well, it turns out that the collection of arrows from one object in a category to another object in the same category is in fact a set.
This set is called a Hom-set, and is very unattractively written Hom(X,Y) is the set of all arrows X → Y.
But Hom-sets are sets, and are therefore objects in the category Set. This means that for an arbitrary category C, I will need a functor C →Set. This guy is called a Hom-functor. It's sole pleasure in life is to take all the arrows X to Y to the appropriate Hom-set.
OK, I am about to introduce one of the key concepts in our theory, but I fear that the length of this post will prevent you from seeing it.
Later. |
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| Guitarist |
| Posted: Sat Jun 14, 2008 5:00 am Post subject: |
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| JaneBennet wrote: |
Whatâs the name for two vector spaces that are âessentially the sameâ (i.e. for which there exists a bijective linear transformation between them)? |
Isomorphic! It is a fundamental theorem that states that any 2 vector spaces of the same dimension are isomorphic.
This is just as well, if you think about it. Recall that the dimension of a vector space is given by the number of basis vectors in the spanning set. Recall also that the SAME vector space can be represented on infinitely many different spanning sets, all of the same cardinality, so we require this space to be isomorphic to itself. That's why I called it a fundamental theorem. |
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| JaneBennet |
| Posted: Sat Jun 14, 2008 5:05 am Post subject: |
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| Guitarist wrote: |
| JaneBennet wrote: |
Whatâs the name for two vector spaces that are âessentially the sameâ (i.e. for which there exists a bijective linear transformation between them)? |
Isomorphic! It is a fundamental theorem that states that any 2 vector spaces of the same dimension are isomorphic.
This is just as well, if you think about it. Recall that the dimension of a vector space is given by the number of basis vectors in the spanning set. Recall also that the SAME vector space can be represented on infinitely many different spanning sets, all of the same cardinality, so we require this space to be isomorphic to itself. That's why I called it a fundamental theorem. |
Thanks. 
| Guitarist wrote: |
| Let C and D be arbitrary categories. The mapping F: C → D is called a functor. |
Shouldnât functors be more than just functions between categories â i.e. they should also preserve the arrows between category objects?
_________________
A problem worthy of attack
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Fact of the day: Old English |
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