Abstract Nonsense

Crushing one theorem at a time

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.

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Motivation

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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” $\mathbf{Ab}$. 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 $\mathbf{Ab}$-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.

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Before we actually define additive categories we prove a somewhat surprising fact about preadditive  categories. Namely, let $\mathcal{C}$ be a preadditvie category and let $x_1,x_2\in\text{obj}(\mathcal{C})$. We say that $z$ is a direct sum of $x_1$ by $x_2$ if there exists morphisms $x_j\xrightarrow{i_j}z$ and $z\xrightarrow{p_j}z$ for $j=1,2$ such that $p_j\circ i_k=\delta_{j,k}$ and $i_1\circ p_1+i_2\circ p_2=1_z$. What we first claim is that a direct sum of $x_1,x_2$ is both a product and coproduct of $x_1,x_2$. Indeed:

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Theorem: Let $\mathcal{C}$ be a preadditive category. Then, if $z$ is a direct sum of $x_1$ by $x_2$ then it is both a product and coproduct.

Proof: We prove that $z$ is a product since the case for coproduct is analogous. Now, what we claim in particular is the guaranteed maps $x_1\xleftarrow{p_1}z\xrightarrow{p_2}x_2$ are actually the required projection maps to make $z$ into a product. Indeed, suppose that we have some object $y$ and morphisms $y\xrightarrow{f_j}x_j$. Define $y\xrightarrow{f}z$ by $f=i_1\circ f_1+i_2\circ f_2$. Note then that

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$p_1\circ f=p_1\circ (i_1\circ f_1+i_2\circ f_2)=(p_1\circ i_1)\circ f_1+(p_1\circ i_2)\circ f_2=1_{x_1}\circ f_1+0\circ f_2=f_1$

$p_2\circ f=p_2\circ (i_1\circ f_1+i_2\circ f_2)=(p_2\circ i_1)\circ f_1+(p_2\circ i_2)\circ f_2=0\circ f_1+1_{x_2}\circ f_2=f_2$

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and moreover, if $g$ is another such map with $p_i\circ f=f_i$ then we see that

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\begin{aligned}f-g &=1_z\circ(f-g)\\ &=(i_1\circ p_1+i_2\circ p_2)\circ(f-g)\\ &=i_1\circ(p_1\circ f-p_2\circ g)+i_2\circ(p_2\circ f- p_2\circ g)\\ &=0\end{aligned}

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and so $f=g$. Since $f_1,f_2$ were arbitrary the fact that $(z,p_1,p_2)$ is a product follows. $\blacksquare$

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Ok, so here is the kind of surprising fact:

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Theorem: Let $\mathcal{C}$ be a preadditive category. Then, the following are equivalent:

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\begin{aligned}&\mathbf{(1)}\quad\mathcal{C}\text{ has finite coproducts}\\ &\mathbf{(2)}\quad\mathcal{C}\text{ has finite product}\\ &\mathbf{(3)}\quad\mathcal{C}\text{ has finite direct sums}\end{aligned}

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Proof: Let’s first prove that $\mathbf{(1)}\Leftrightarrow\mathbf{(3)}$. We have already proven that $\mathbf{(3)}\implies \mathbf{(1)}$ and so it suffices to prove that $\mathbf{(1)}\implies \mathbf{(3)}$. To do this take two arbitrary $x_1,x_2\in\text{obj}(\mathcal{C})$. We claim that their coproduct $(x_1\sqcup x_2,i_1,i_2)$ is actually a direct sum with $p_i$ the unique maps $p_i:x_1\sqcup x_2\to x_i$ such that $p_i\circ i_j=\delta_{i,j}$ (which we know exist by definition of the coproduct). Well, evidently we have that the $p_i$ satisfy two of the necessary conditions to make $(x_1\sqcup x_2,i_1,i_2,p_1,p_2)$ a direct sum–it remains to check that $i_1\circ p_1+i_2\circ p_2=1_{x_1\sqcup x_2}$. That said, it’s easy to see that $i_1\circ p_1+i_2\circ p_2$ is a morphism $x_1\sqcup x_2\to x_1\sqcup x_2$ which, as can easily be checked, satisfies

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$(i_1\circ p_1+i_2\circ p_2)\circ i_1=i_1$

$(i_1\circ p_2+i_2\circ p_2)\circ i_2=i_2$

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and since $1_{x_1\sqcup x_2}$ also postcomposes with the $i_j$ to the same result, we may conclude by uniqueness that $i_1\circ p_1+i_2\circ p_2=1_{x_1\sqcup x_2}$ as desired. Thus, $(1_{x_1\sqcup x_2},i_1,i_2,p_1,p_2)$ is a direct sum.

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We leave $\mathbf{(2)}\Leftrightarrow\mathbf{(3)}$ to the reader since it’s very similar. $\blacksquare$

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This pecularity allows us to define an additive category $\mathcal{C}$ to be a preadditive category that satisfies any of the three equivalent properties laid out in the previous theorem. Moreover, we see that if $x_1,x_2$ are objects in the additive category $\mathcal{C}$, their product, coproduct, and direct sum are all naturally isomorphic and thus we use the symbol $x_2\oplus x_2$ to denote all three.

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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 $(z,i_1,\cdots,i_n,p_1,\cdots,p_n)$ with $p_i\circ i_j=\delta_{i,j}$ and $\displaystyle \sum_{i,j=1}^{n}i_j\circ p_i=1_z$.

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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] 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.

April 2, 2012 -