Abelian Categories (Pt. II)
Point of Post: This is a continuation of this post.
Ok, so we have twothirds of the first isomorphism theorem ingredients down. What is the third? Well, while the first isomorphism theorem is often stated “if is a morphism then ” the real statement is that the natural map given by is an isomorphism–much more strict than the statement that the two objects are isomorphic. Ok, so the first isomorphism theorem that we’d like to state should involve the third ingredient, which is a natural map between and . What exactly does natural mean here? For right now, take natural to mean “independent of actual identity of set/structure/etc.” In other words, the map between two objects should come entirely from what the objects are (various kernels and cokernels) and have nothing to do with what they might be in specific cases (modules, sets, etc.) So, we now wish to get a map just by fiddling around with the definitions of kernel and cokernel.
To do this we start with the diagram that we know. Namely, let’s say we are in some preabelian category with objects and a morphism . We then obviously can append and to get the following diagram
We can then add in our new constructions of and to get
Now, since we get by the definition of a morphism making the following diagram commute
Noting then that and the fact that is epi (since all cokernel maps are epi) we may conclude that . Using then the definition of we get an arrow making the following diagram commute
The beauty of the above is that since we only used the properties of kernels and cokernels, we know that each step the constructed map is unique. Thus, the above diagram can be thought of in a unique and natural way. Thus, we shall never actually explicitly name the map since there is really only one natural map it could be .
One can check that if all of this takes place in then the map is the normal map . Thus, the final piece of the “first isomorphism theorem” is laid snuggly in place. Namely, the first isomorphism theorem can be stated, glibly, as –that the natural map is an isomorphism.
Thus, we are finally prepared to formally define the white whale. An abelian category is a preabelian category such that for all diagrams in .
What has always startled me was that the above is actually NOT the standard definition of abelian category. Indeed, other authors opt to take (to me) more opaque definitions. For example, we have the following theorem:
Theorem: Let be a preabelian category. Then, is abelian if and only if every monomorphism is the kernel of some morphism and every epimorphism is the cokernel of some morphism.
While we shall not prove this, we shall use it freely (do the proof–it’ll be good for your soul).
I’d like to wrap this post up by mentioning that famous theorem I was talking about in the introduction. Namely, the Freyd(Mitchell) Embedding theorem:
Theorem(FreydMitchell Embedding Theorem): Let be a small abelian category. Then, there exists a ring and a full faithful embedding .
The proof of this is fairly involved and fairly elusive. An honest to god proof of it can be found in [4] around page 150 and somewhere in [5] (the books being respectively by Freyd and Mitchell).
The most imporant corollary of this theorem is the following metaprinciple:
Metaprinciple: Any statement about abelian categories which is provable in and only involves finitely many objects and arrows is true in any abelian category.
Roughly the reason for this, is that we can just fully and faithfully embed any diagram (which can be put into a small abelian subcategory) into where the statement is true. Because of this metaprinciple there will often times be proof where we only actually prove the result for (e.g. the snake lemma).
References:
[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: SpringerVerlag, 1994. Print.
[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.
[3] Berrick, A. J., and M. E. Keating. Categories and Modules with Ktheory in View. Cambridge, UK: Cambridge UP, 2000. Print.
[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.
[5] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.
[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.
April 2, 2012  Posted by Alex Youcis  Algebra, Category Theory  Abelian Categories, Additive Categories, Algebra, Category Theory, Embedding Theory, First Isomorphism Theorem, Metaprinciple, Motivation
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