Abstract Nonsense

Crushing one theorem at a time

Ab-categories and Preadditive Categories (Pt. III)


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

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Preadditive Categories

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It’s evident from the above that \mathbf{Ab}-categories interact very nicely with zero objects. Moreover, it’s clear in most contexts \mathbf{Ab}-categories are going to have zero objects. Thus, it often seems useful to assume that \mathbf{Ab}-categories have zero objects, in which case we call such a category preadditive. What is an example of an \mathbf{Ab}-category which isn’t preadditive? Well, from the above it’s easy to see that for a given unital ring R one has that \mathcal{C}_R (an \mathbf{Ab}-category) is going to be preadditive if and only if R=0 (this follows from the above proven fact that \text{End}_R(R,R)=R must be trivial). For the convenience of future proofs we make the following observation about the equivalent forms of a zero object in a \mathbf{Ab}-category:

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Theorem: Let \mathcal{C} be an \mathbf{Ab}-category. Then, for an object z in \mathcal{C} the following are equivalent

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\begin{aligned}&\mathbf{(1)}\quad 0_{z,z}=1_z\\&\mathbf{(2)}\quad \text{End}(z)\textit{ is trivial}\\&\mathbf{(3)}\quad z\textit{ is initial }\\&\mathbf{(4)}\quad z\textit{ is terminal}\\ &\mathbf{(5)}\quad z\textit{ is a zero object}\end{aligned}

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Proof: Basically the entire theorem has already been proven except for the parts pertaining to \mathbf{(3)} and \mathbf{(4)}. But, the idea behind them is simple, if z is terminal or initial then there is a unique morphism z\to z and so \text{End}(z) must be trivial. \blacksquare

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

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

4 Comments »

  1. […] ), called an -map. The set of all -maps is denoted .  It’s not hard to see that is a preadditive category with the usual sum of functions and the zero object being the zero -bimodule. Obviously the […]

    Pingback by The Construction of the Tensor Product of Modules (Pt. II) « Abstract Nonsense | January 4, 2012 | Reply

  2. […] described above) is additive. But, this follows immediately since if are arrows then since , as previously discussed, is the additive identity of […]

    Pingback by Functor Categories (Pt. II) « Abstract Nonsense | January 7, 2012 | Reply

  3. […] what are they? Well, we have slowly been building up to this point, defining -categories, and then preadditive categories, and then finally additive categories. So, what’s the next step? Depending on how finely we […]

    Pingback by Abelian Categories (Pt. I) « Abstract Nonsense | April 2, 2012 | Reply

  4. […] like to do is pass from what we have now, an -category, to the next step up the ladder–preadditive categories. Of course, this is equivalent to identifying a zero object for our category. Of course, […]

    Pingback by Chain Complexes « Abstract Nonsense | April 3, 2012 | Reply


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