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Math Help - Category Definition

  1. #1
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    Category Definition

    Can someone define for me using sets what a category is. The sites I tried seem to avoid using the word "set". Is that intentional?
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    Quote Originally Posted by ThePerfectHacker View Post
    Is that intentional?
    Yes, they're trying to annoy you
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    Yes, it is intentional. By and large, categories cannot exist as sets, but only as proper classes. Consider for example the category of all fields. There is essentially no way, using the standard axioms of set theory, to build a set whose elements are representatives of every possible isomorphism class of fields.

    I think you and I ran into this problem during a different discussion... you were inquiring, if I recall, as to whether the algebraic closure of a field was necessarily a maximal extension of that field. You correctly deduced that, if there were such a thing as "the set of all extension fields of a field F", then by Zorn's Lemma, we would have a maximal extension for F. But there is no such thing... we can always adjoin indeterminates to a field to obtain a bigger one. So we must conclude (if we believe a priori that the axioms of set theory are consistent) that there is no such set. We must then further conclude that there is no such thing as "the set of all fields".

    Abstractly, a category is a collection (proper class) of objects, together with a bunch of "morphisms" between the objects, that behave according to certain rules. There is nothing in the definition that requires the morphisms to be functions, nor the objects themselves to be sets, but this is usually the case for most concrete applications. Category theory is an active field of study to be sure, but for most practicing mathematicians (or at least myself), it merely provides a convenient framework and language to speak of the relationships between certain objects in the same category.

    Take the category of groups. The natural choice for the "morphisms" would be the homomorphisms that exist between groups. Categories always admit a certain 'multiplication' of morphisms, and the natural choice for this multiplication would be function composition. You also have the concepts of "equivalence", "products", "free objects", "universal objects", "terminal objects", etc. These would happen to correspond to, respectively, "isomorphic", "direct product", "free group (on a given set of generators)", "the trivial group" and, again, "the trivial group".

    About the only book I can recommend is Hungerford's "Algebra", and even there he only has a couple of sections, but it's enough to give you the flavor of the thing. I would make a point to ask your teacher exactly how in depth he expects you to know this stuff. Like I said, it's usually enough to understand the basic concept and definitions, and to see how the same relationships between objects in mathematics seem to come up over and over again... homomorphisms between groups, linear maps between vector spaces, continuous maps between topological spaces, etc. So unless he says otherwise, I wouldn't get too carried away with it (unless it turns out you really like it of course). Later
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    Quote Originally Posted by BubbleBrain_103 View Post
    I think you and I ran into this problem during a different discussion... you were inquiring, if I recall, as to whether the algebraic closure of a field was necessarily a maximal extension of that field.
    Ah! That was what I was thinking of when you said that. I remember that day we were discussing it, what beauty.

    Abstractly, a category is a collection (proper class) of objects, together with a bunch of "morphisms" between the objects, that behave according to certain rules.
    So, basically a category is a set that can violate certain ZFC axioms and hence be a proper class. So instead of saying set we say category to indicate that it can be non-containable. For example, what you said.... The set of all groups cannot be treated like a set. However, we can speak about the category of all groups. And instead of saying the elements of the set, we say the objects of the set. And when you say "morphism" you mean an analogue of a binary operation. (I am just so curios to know, can you have the category of all categories .... or is that category so so so large that it violates the Category axioms?)

    I was speaking with some professor that I like to discuss math with and one of the thing I mentioned was a category after I seen your explaination but did not have time to read it. And he said to me that when he was much younger he used to play around with sets and many times he would violate the axioms. So he ran to one of his professors who was one of the developers of category theory and asked him why does this not work. The professor always used to respond to him, "do not worry this all makes sense in category theory".
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