# Abstract Nonsense

## Abelian Categories (Pt. II)

Point of Post: This is a continuation of this post.

Ok, so we have two-thirds of the first isomorphism theorem ingredients down. What is the third? Well, while the first isomorphism theorem is often stated “if $f:X\to Y$ is a morphism then $X/\ker f\cong\text{im }f$” the real statement is that the natural map $X/\ker f\to\text{im }f$ given by $x+\ker f\mapsto f(x)$ 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 $\text{im }f$ and $\text{coim }f$. 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 $\text{coim }f\to\text{im }f$ just by fiddling around with the definitions of kernel and cokernel.

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To do this we start with the diagram that we know. Namely, let’s say we are in some preabelian category $\mathcal{C}$ with objects $X,Y$ and a morphism $X\xrightarrow{f}Y$. We then obviously can append $\ker f$ and $\text{coker }f$ to get the following diagram

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$\ker f\xrightarrow{h}X\xrightarrow{f}Y\xrightarrow{s}\text{coker }f$

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We can then add in our new constructions of $\text{coim }f$ and $\text{im }f$ to get

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$\begin{array}{ccccccc}\ker f & \xrightarrow{s} & X & \xrightarrow{f} & Y & \xrightarrow{s} & \text{coker }f\\ & & ^\alpha\big\downarrow & & \big\uparrow^\beta & &\\ & & \text{coim }f & & \text{im }f & & \end{array}$

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Now, since $f\circ h=0$ we get by the definition of $\text{coker}(\ker f\xrightarrow{h}X)$ a morphism  $\widetilde{f}$ making the following diagram commute

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$\begin{array}{ccccccc}\ker f & \xrightarrow{s} & X & \xrightarrow{f} & Y & \xrightarrow{s} & \text{coker }f\\ & & ^\alpha\big\downarrow & \overset{\widetilde{f}}{\nearrow} & \big\uparrow^\beta & &\\ & & \text{coim }f & & \text{im }f & & \end{array}$

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Noting then that $s\circ\widetilde{f}\circ\alpha=s\circ f=0$ and the fact that $\alpha$ is epi (since all cokernel maps are epi) we may conclude that $s\circ\widetilde{f}=0$. Using then the definition of $\ker(Y\xrightarrow{s}\text{coker }f)$ we get an arrow $u$ making the following diagram commute

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$\begin{array}{ccccccc}\ker f & \xrightarrow{s} & X & \xrightarrow{f} & Y & \xrightarrow{s} & \text{coker }f\\ & & ^\alpha\big\downarrow & \overset{\widetilde{f}}{\nearrow} & \big\uparrow^\beta & &\\ & & \text{coim }f & \xrightarrow{u} & \text{im }f & & \end{array}$

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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 $\text{coim }f\to\text{im }f$ since there is really only one natural map it could be .

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One can check that if all of this takes place in $R\text{-}\mathbf{Mod}$ then the map $\text{coim }f\to\text{im }f$ is the normal map $x+\ker f\mapsto f(x)$. Thus, the final piece of the “first isomorphism theorem” is laid snuggly in place. Namely, the first isomorphism theorem can be stated, glibly, as $\text{coim }f\xrightarrow{\approx}\text{im }f$–that the natural map $\text{coim }f\to\text{im }f$ is an isomorphism.

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Thus, we are finally prepared to formally define the white whale. An abelian category $\mathscr{A}$ is a preabelian category such that $\text{coim }f\xrightarrow{\approx}\text{im }f$ for all diagrams $X\xrightarrow{f}Y$ in $\mathscr{A}$.

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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:

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Theorem: Let $\mathcal{C}$ be a preabelian category. Then, $\mathcal{C}$ is abelian if and only if every monomorphism is the kernel of some morphism and every epimorphism is the cokernel of some morphism.

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While we shall not prove this, we shall use it freely (do the proof–it’ll be good for your soul).

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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:

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Theorem(Freyd-Mitchell Embedding Theorem): Let $\mathscr{A}$ be a small abelian category. Then, there exists a ring $R$ and a full faithful embedding $F:\mathscr{A}\to R\text{-}\mathbf{Mod}$.

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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).

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The most imporant corollary of this theorem is the following metaprinciple:

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Metaprinciple: Any statement about abelian categories which is provable in $\mathbf{Ab}$ and only involves finitely many objects and arrows is true in any abelian category.

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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 $\mathbf{Ab}$ where the statement is true. Because of this metaprinciple there will often times be proof where we only actually prove the result for $\mathbf{Ab}$ (e.g. the snake lemma).

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References:

[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 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 K-theory 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 -