## Additive Categories (Pt. II)

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

**Matrix Representation of Morphisms**

We next discuss how we can represent a morphism between direct sums as a “matrix”. The key observation is this: given any four morphisms for there exists a unique morphism such that . Indeed, we merely let

Clearly this morphism satisfies the desired properties with respect to composition with the and and is unique for obvious reasons. Conversely, any morphism can be decomposed in this way. Thus, we can identify morphisms with matrices. Explicitly, the morphism is associated to the matrix where . More generally, given objects and we can identify morphisms with size matrices with . Once again, we can recover from via the formula

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

Now, what would make our day? Well, the first thing we could hope for is that 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 (which I won’t make an effort to distinguish between ‘s originating from different pairs of direct sums) distributed over the other operation in –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

what we’d like to check is that . But, this is just simple. For example, in the first entry of we get

where we used the fact that . Of course, the other entries are done entirely analgously and thus we gain the desired result that respects the multiplicative structure of . Of course, also respects the multiplicative structure and additive structure.

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.

**Additive Functors**

Additive functors definitely take on new life when paired with additive categories instead of preadditive or -categories. Indeed, it’s unsurprsing considering the “additive” formulation of direct sums that additive functors should preserve coproducts. Explicitly, note that if is a coproduct in and is an additive functor then is a direct sum of and . Indeed, using the fact that preserves sums, zeros, identities, and compositions we can check that and from where the conclusion follows. Thus, (or equality, if one so wanted).

The surprising thing is that, in fact, this characterizes additive functors for additive categories. Namely, let be a functor, where and are additive categories. Then, 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.

**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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