Abstract Nonsense

Crushing one theorem at a time

Additive Categories (Pt. II)


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

Matrix Representation of Morphisms

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We next discuss how we can represent a morphism between direct sums as a “matrix”. The key observation is this: given any four  morphisms x_i\xrightarrow{f_{i,j}}y_j for j=1,2 there exists a unique morphism x_1\oplus x_2\xrightarrow{f}y_1\oplus y_2 such that p_{y_i}\circ f \circ i_{x_j}=f_{i,j}. Indeed, we merely let

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\displaystyle f=\sum_{i=1}^{2}\sum_{j=1}^{2}i_{y_i}\circ f_{i,j}\circ p_{x_i}

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Clearly this morphism satisfies the desired properties with respect to composition with the p_{y_j} and i_{x_j} and is unique for obvious reasons. Conversely, any morphism x_1\oplus x_2\to y_1\oplus y_2 can be decomposed in this way. Thus, we can identify morphisms x_1\oplus x_2\to y_1\oplus y_2 with 2\times 2 matrices. Explicitly, the morphism x_1\oplus x_2\to y_1\oplus y_2 is associated to the matrix \left(\begin{smallmatrix}f_{1,1} & f_{1,2}\\ f_{2,1} & f_{2,2}\end{smallmatrix}\right) where f_{i,j}=p_{y_i}\circ f\circ i_{x_j}. More generally, given objects x_1,\cdots,x_n and y_1,\cdots,y_m we can identify morphisms x_1\oplus\cdots\oplus x_n\to y_1\oplus\cdots\oplus y_m with m\times n size matrices M=(f_{i,j}) with f_{i,j}=p_{y_i}\circ f\circ i_{x_j}. Once again, we can recover f from M via the formula

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\displaystyle f=\sum_{i=1}^{m}\sum_{j=1}^{n}i_{y_i}\circ f_{i,j}\circ p_{x_j}

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Ok, so this is pretty cool way to rename/index morphisms on direct sums, yeah? That said, like any indexing systems worth its salt, there should be some utility of the indexing that goes beyond just naming. Indeed, what we have done is constructed inverse maps

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\text{Hom}_{\mathcal{C}}(x_1\oplus\cdots\oplus x_n,y_1\oplus\cdots\oplus y_m)\overset{\displaystyle \overset{M}{\longrightarrow}}{\displaystyle \underset{N}{\longleftarrow}}\left(\begin{array}{ccc}\text{Hom}_\mathcal{C}(x_1,y_1) & \cdots & \text{Hom}_\mathcal{C}(x_n,y_1)\\ \vdots & \ddots & \vdots\\ \text{Hom}_\mathcal{C}(x_1,y_m) & \cdots & \text{Hom}_\mathcal{C}(x_n,y_m)\end{array}\right)

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Now, what would make our day? Well, the first thing we could hope for is that M,N are group homomorphisms (since both sides are groups, of course). But, this is really quite obvious, right (i.e. I’m lazy and don’t want to write it out–I’m sure you’ll have no trouble showing it). That said, what would be really great is if this function M (which I won’t make an effort to distinguish between M‘s originating from different pairs of direct sums) distributed over the other operation in \mathcal{C}–namely, composition. Well, as all things in basic category theory that should be true, are true, it is true. Let’s check it. Namely, suppose that we have

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x_1\oplus\cdots\oplus x_n\overset{f}{\longrightarrow}y_1\oplus\cdots\oplus y_m\overset{g}{\longrightarrow}z_1\oplus\cdots\oplus z_k

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what we’d like to check is that M(g\circ f)=M(g)M(f). But, this is just simple. For example, in the first entry of M(g)M(f) we get

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\begin{aligned}\sum_{j=1}^{m}(p_{z_1}\circ g\circ i_{y_j})\circ(p_{y_j}\circ f\circ i_{x_1}) &=p_{z_1}\circ g\circ\left(\sum_{j=1}^{m}i_{y_j}\circ p_{y_j}\right)\circ f\circ i_{x_1}\\ &=p_{z_1}\circ (g\circ f)\circ i_{x_1}\end{aligned}

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where we used the fact that \displaystyle \sum_{j=1}^{m}i_{y_j}\circ p_{y_j}=1_{y_1\oplus\cdots\oplus y_m}. Of course, the other entries are done entirely analgously and thus we gain the desired result that M respects the multiplicative structure of \mathcal{C}. Of course, N also respects the multiplicative structure and additive structure.

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The above analysis makes it very convenient for us to describe morphisms of direct sums in additive categories as matrices since they are a) easy to define and b) easy to compute with.

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Additive Functors

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Additive functors definitely take on new life when paired with additive categories instead of preadditive or \mathbf{Ab}-categories. Indeed, it’s unsurprsing considering the “additive” formulation of direct sums that additive functors should preserve coproducts. Explicitly, note that if (x_1\oplus x_2,i_1,i_2,p_1,p_2) is a coproduct in \mathcal{A} and F:\mathcal{A}\to\mathcal{B} is an additive functor then (F(x_1\oplus x_2),F(i_1),F(i_2),F(p_1),F(p_2)) is a direct sum of F(x_1) and F(x_2). Indeed, using the fact that F preserves sums, zeros, identities, and compositions we can check that F(p_i)\circ F(i_j)=\delta_{i,j} and F(i_1)\circ F(p_1)+F(i_2)\circ F(p_2)=1_{F(x_1\oplus x_2)} from where the conclusion follows. Thus, F(x_1\oplus x_2)\cong F(x_1)\oplus F(x_2) (or equality, if one so wanted).

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The surprising thing is that, in fact, this characterizes additive functors for additive categories. Namely, let F:\mathcal{A}\to\mathcal{B} be a functor, where \mathcal{A} and \mathcal{B} are additive categories. Then, F is additive if and only if it preserves zeros and direct sums. I won’t prove this here, but it can be found in any good textbook on category theory/homological algebra.

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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.

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April 2, 2012 - Posted by | Algebra, Category Theory | , , , , , , ,

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