## Ab-categories and Preadditive Categories (Pt. III)

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

*Preadditive Categories*

It’s evident from the above that -categories interact very nicely with zero objects. Moreover, it’s clear in most contexts -categories are going to have zero objects. Thus, it often seems useful to assume that -categories have zero objects, in which case we call such a category *preadditive*. What is an example of an -category which isn’t preadditive? Well, from the above it’s easy to see that for a given unital ring one has that (an -category) is going to be preadditive if and only if (this follows from the above proven fact that must be trivial). For the convenience of future proofs we make the following observation about the equivalent forms of a zero object in a -category:

**Theorem: ***Let be an -category. Then, for an object in the following are equivalent*

**Proof: **Basically the entire theorem has already been proven except for the parts pertaining to and . But, the idea behind them is simple, if is terminal or initial then there is a unique morphism and so must be trivial.

**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] Rotman, Joseph J. *Introduction to Homological Algebra*. Springer-Verlag. Print.

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