## Additive Categories (Pt. I)

**Point of Post: **In this post we discuss the notion of additive categories, including the fact that finite products and coproducts are isomorphic, as well as the fact that functor is additive if and only if preserves zero objects and coproducts.

**Motivation**

We are (somewhat interminably) making our way towards discussing abelian categories which are the categories general enough to do homological algebra on–they are the ones that “resemble” . We have already discussed some of the approximating classes of categories and in this post we go to the next level of specificity after preadditive categories–additive categories. We are not adding much to get from pre-additive to additive, really we are just insisting that the category have finite products and coproducts. That said, some unexpectedly nice things happen when we start to think about products and coproducts in categories where we have an -feel. In particular we shall see that for additive categories finite products and coproducts are actually the same (known then as ‘biproducts’) which is certainly the case in our favorite additive categories–categories of modules. Moreover, we shall see that being able to discuss products makes our life easier when we start considering additive functors for, as we shall see, additive functors are nothing more than functors preserving biproducts and zero objects. All of this stems from a characterization of coproducts (and by extension, products) involving the additive structure between the Hom sets.

**Additive Categories**

Before we actually define additive categories we prove a somewhat surprising fact about preadditive categories. Namely, let be a preadditvie category and let . We say that is a *direct sum *of by if there exists morphisms and for such that and . What we first claim is that a direct sum of is both a product and coproduct of . Indeed:

**Theorem: ***Let be a preadditive category. Then, if is a direct sum of by then it is both a product and coproduct.*

**Proof: **We prove that is a product since the case for coproduct is analogous. Now, what we claim in particular is the guaranteed maps are actually the required projection maps to make into a product. Indeed, suppose that we have some object and morphisms . Define by . Note then that

and moreover, if is another such map with then we see that

and so . Since were arbitrary the fact that is a product follows.

Ok, so here is the kind of surprising fact:

**Theorem: ***Let be a preadditive category. Then, the following are equivalent:*

**Proof: **Let’s first prove that . We have already proven that and so it suffices to prove that . To do this take two arbitrary . We claim that their coproduct is actually a direct sum with the unique maps such that (which we know exist by definition of the coproduct). Well, evidently we have that the satisfy two of the necessary conditions to make a direct sum–it remains to check that . That said, it’s easy to see that is a morphism which, as can easily be checked, satisfies

$latex\ text{ }$

and since also postcomposes with the to the same result, we may conclude by uniqueness that as desired. Thus, is a direct sum.

We leave to the reader since it’s very similar.

This pecularity allows us to define an *additive category *to be a preadditive category that satisfies any of the three equivalent properties laid out in the previous theorem. Moreover, we see that if are objects in the additive category , their product, coproduct, and direct sum are all naturally isomorphic and thus we use the symbol to denote all three.

Of course, in a preadditive category we can define the direct sum of finitely many objects, and considering that they are just products/coproducts they enjoy all of the associativity and commutativity. Moreover, it’s not hard to see that we can characterize direct sums in the form with and .

**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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