# Abstract Nonsense

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

Point of Post: In this post we define the notion of $\mathbf{Ab}$-categories and preadditive categories, give some simple theorems when an $\mathbf{Ab}$-category is a preadditive category, and give some examples of both. We also discuss the notion of additive functors.

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Motivation

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We are about to discuss the first step in our journey to defining precisely what an abelian category is. As can be guessed from our current lack of categorical machinery it’s not hard to guess that whatever $\mathbf{Ab}$-categories and preadditive categories are, they aren’t ‘too in-depth’. In fact, the idea behind both types of categories is quite natural and shows up very often in mathematics, most notably in modules theory and homological algebra. So, what is an $\mathbf{Ab}$-category? A fancy way of answering this question is “it’s a category enriched over $\mathbf{Ab}$“, but what does this mean? Roughly what it’s saying is that an $\mathbf{Ab}$-category is a category which is closely linked with $\mathbf{Ab}$, in particular it’s Hom sets are elements of $\mathbf{Ab}$. Moreover, we’d like to require that the composition in each of these Hom sets respects this abelian group structure, in the sense that it left and right distributes (e.g. $f\circ (g+h)=f\circ g+f\circ h$). It should now be clear that such categories are of interest to us, in that they pop up very often. For example, given any ring $R$ we have already proven that $R\text{-}\mathbf{Mod}$ is an $\mathbf{Ab}$-category, and of course any full subcategory of $R\text{-}\mathbf{Mod}$ will be as well. Moreover, the axioms of the Hom sets of an $\mathbf{Ab}$-category should look pretty familiar, because just as a category with one object is a monoid an $\mathbf{Ab}$-category with one object is just a ring. To go from a $\mathbf{Ab}$-category to a preadditive category we must merely require that the category has a zero object, as do all the above examples of $\mathbf{Ab}$-categories.

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$\mathbf{Ab}$-categories, Preadditive Categories, and Additive Functors

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Let $\mathcal{C}$ be a category. Then, if $+=\{+_{(x,y)}\}$ is a set of binary operations $+_{(x,y)}:\text{Hom}(x,y)\times\text{Hom}(x,y)\to\text{Hom}(x,y)$ such that $\left(\text{Hom}(x,y),+_{(x,y)}\right)$ is an abelian group, and such that if $x\overset{\displaystyle\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}y\xrightarrow{h}z$ and $x\xrightarrow{s}y\overset{\displaystyle\overset{t}{\longrightarrow}}{\underset{u}{\longrightarrow}}z$ then $h\circ(f+_{(x,y)}g)=(h\circ f)+_{(x,z)}(h\circ g)$ and $(t+_{(y,z)}u)\circ s=(t\circ s)+_{(x,z)}(u\circ s)$, then we call the ordered pair $\left(\mathcal{C},+\right)$ an $\mathbf{Ab}$category.

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Of course, in practice, we shall not differentiate between $+$ and the individual binary operations $+_{(x,y)}$, and so we shall write, for example, just $f+g$ (this is in agreement with our writing of $g\circ f$ instead of $g\circ_{(x,y)}f$).

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Let’s now look at some examples of $\mathbf{Ab}$-categories:

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Let $R$ be any ring, then we have proven (as discussed in the motivation) that $R\text{-}\mathbf{Mod}$ is an $\mathbf{Ab}$-category, from where we recover that (as should be) $\mathbf{Ab}\cong\mathbb{Z}\text{-}\mathbf{Mod}$ is an $\mathbf{Ab}$-category.

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What we’d now like to justify is our claim that unital rings are nothing more than $\mathbf{Ab}$-categories with one object. Indeed, suppose $R$ is a ring and let $\mathcal{C}_R$ be the category with one object, $R$, and $\text{Hom}_{\mathcal{C}_R}(R,R)=R$, and composition is just $R$-multiplication. Note then that the Hom sets of $\mathcal{C}_R$, namely $R$, are naturally imbued with an abelian group structure, the one underlying the ring structure of $R$, and moreover the left and right distributivity axioms for composition/addition follow from the left right distributivity axioms for addition/$R$-multiplication. Conversely, suppose that we are given an $\mathbf{Ab}$-category $\mathcal{C}$ with one object $x$–then everything follows by considering $\text{End}(x)=\text{Hom}(x,x)$ and the following theorem:

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Theorem: Let $\left(\mathcal{C},+\right)$ be an $\mathbf{Ab}$-category and $x$ an object of $\mathcal{C}$. Then, $\text{End}(x)$ is a unital ring with multiplication given by composition, and addition by $+$.

Proof:The fact that $\left(\text{End}(x),+\right)$ is an abelian group and $\left(\text{End}(x),\circ\right)$ is a monoid, follow immediately from the definition of an $\mathbf{Ab}$-category. Moreover, the distributitvity between $+$ and $\circ$ is also in the definition. $\blacksquare$

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Given a ring $R$ and a preordered set $\left(\mathcal{A},+\right)$, the category of directed systems $\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)$ is an $\mathbf{Ab}$-category when we define the sum of two morphisms $\{w_\alpha\}$ and $\{v_\alpha\}$, from $(\{M_\alpha\},\{f_{\alpha,\beta}\})$ to $(\{N_\alpha\},\{g_{\alpha,\beta}\})$, to be $\{w_\alpha+v_\alpha\}$ where the sum now takes place in $\text{Hom}_R(M_\alpha,N_\alpha)$. To see, first of all, that $\{w_\alpha+v_\alpha\}$, still a morphism we merely note that

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\begin{aligned} g_{\alpha,\beta}\circ(w_\alpha+v_\alpha) &=(g_{\alpha,\beta}\circ w_\alpha)+(g_{\alpha,\beta}\circ w_\alpha)\\ &=(w_\beta\circ f_{\alpha,\beta})+(v_\beta\circ f_{\alpha,\beta})\\ &=(w_\beta+v_\beta)\circ f_{\alpha,\beta}\end{aligned}

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The rest of the axioms for $\mathbf{DS}_\mathcal{A}\left(R\text{-}\mathbf{Mod}\right)$ to be an $\mathbf{Ab}$-category follow similarly by applying “coordinate-wise” the fact that $R\text{-}\mathbf{Mod}$ is an $\mathbf{Ab}$-category (in fact, this greatly hints at, and is a corollary of, the more general fact that if $\mathcal{S}$ is a small category and $\mathcal{C}$ is an $\mathbf{Ab}$-category then the functor category $\mathcal{C}^\mathcal{S}$ is naturally an $\mathbf{Ab}$-category with this same “coordinate-wise” operations).

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Now, we note that there is an interesting, but somewhat semantical, fact about groups that keeps a lot of categories from even hoping to be an $\mathbf{Ab}$-category. In particular, since given any two objects $x,y$ in some $\mathbf{Ab}$-category $\mathcal{C}$ we have that $\text{Hom}(x,y)$ is a group, we know that $\text{Hom}(x,y)\ne\varnothing$ (the identity element!). Thus, categories that have empty Hom sets can’t be made into $\mathbf{Ab}$-categories under any addition operation. For example, this shows that the category $\mathbf{Ring}$ and the category $\mathbf{Top}$ cannot be made into $\mathbf{Ab}$-categories (see the discussion at the bottom of this page).

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Speaking of zero objects, the following two theorem tells us that agreeing with our intuition the zero morphism $x\xrightarrow{0_{x,y}}y$ must be the additive identity for $\text{Hom}(x,y)$. Indeed:

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Theorem: Let $\mathcal{C}$ be an additive category and $x,y,z$ objects in $\mathcal{C}$. Then, if $0_{x,y}$ and $0_{y,z}$ denote the additive identities of $\text{Hom}(x,y)$ and $\text{Hom}(y,z)$ respectively then $o_{y,z}\circ o_{x,y}=0_{x,z}$.

Proof: Note that $0_{x,z}=0_{y,z}\circ0_{x,y}-0_{y,z}\circ0_{x,y}=0_{y,z}\circ(0_{x,y}-0_{x,y})=0_{y,z}\circ0_{x,y}$. $\blacksquare$

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Theorem: Let $\mathcal{C}$ be an additive category and suppose that $\mathcal{C}$ has a zero object, $0$. Then, for any objects $x,y$ in $\mathcal{C}$ we have that the additive identity of $\text{Hom}(x,y)$ is the zero morphism $x\xrightarrow{0_{x\to y}}y$ (where we only write $x\to y$ to distinguish between the additive identity $0_{x,y}$ o $\text{Hom}(x,y)$ although, ultimately, they are equal).

Proof: We know that if $x\xrightarrow{f}0$ and $0\xrightarrow{g}y$ are the unique morphisms in those directions then $0_{x\to y}=g\circ f$. But, note that since $\text{Hom}(x,0)$ and $\text{Hom}(0,y)$ are singletons that it must be the case that $f=0_{x,0}$ and $g=0_{0,y}$ and so $0_{x\to y}=g\circ f=0_{0,y}\circ0_{x,0}=0_{x,y}$. The conclusion follows. $\blacksquare$

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We now have the following general facts about how to create more $\mathbf{Ab}$-categories:

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Theorem: Let $\left(\mathcal{C},+\right)$ be an $\mathbf{Ab}$-category. Then, the opposite category $\left(\mathcal{C}^\text{op},+'\right)$ with $+'_{(x,y)}=+_{(y,x)}$ is an $\mathbf{Ab}$-category

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Theorem: Let $\left(\mathcal{C},+_1\right)$ and $\left(\mathcal{D},+_2\right)$ be two $\mathbf{Ab}$-categories. Then, $\left(\mathcal{C}\times\mathcal{D},+\right)$ is an $\mathbf{Ab}$-category where $(f,g)+(h,u)=(f+_1h,g+_2u)$.

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Theorem: Let $\left(\mathcal{C},+\right)$ be an $\mathbf{Ab}$-category and $\mathcal{D}$ a full subcategory of $\mathcal{C}$, then $\left(\mathcal{D},+\right)$ is an $\mathbf{Ab}$-category.

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All of these theorems are trivial to prove.

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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.

January 2, 2012 -

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5. […] 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). […]

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