# Abstract Nonsense

## Chain Complexes

Point of Post: In this post we discuss the notion of chain complexes in additive and abelian categories

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Motivation

Now that we have finally defined abelian categories we can start discussing the objects that interesting us–namely chain complexes. Chain complexes should be very familiar to anyone who has done the slightest bit of algebra or topology should be well acquainted with chain complexes. Indeed, chain complexes show up in the form often times when one is able to describe an object $X$ as being “put together” from objects $A$ and $B$. For example, the existence of a short exact sequence $0\to A\to X\to B\to 0$ in $\mathbf{Ab}$  indicates that $X$ is some kind of “combination” of $A$ and $B$ in such a way that $X/A\cong B$–in the best case scenario $X\cong A\oplus B$.  Thus, a common technique to classify all algebraic objects satisfying property $P$ is to prove that all such objects can be put in a short exact sequence $0\to A_0\to X\to B_0\to 0$ where $A_0,B_0$ are known, and thus we have reduced our questioning to find exactly which $X$ can fit into that sequence. In general, putting $X$ into a long complex $\cdots \to A\to X\to B\to\cdots$ indicates that we have decomposed $X$, and that how “nice” this decomposition is depends on how the rest of the chain plays out.

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Chain complexes come up in topology in a lot of contexts, probably the one which is closest to the surface being the singular chain complex coming from the singular homology of a space. That said, they also come up when doing slightly more exotic constructions like the DeRham cohomology of a space.

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Regardless, in all of the “basic” mathematics that we have seen chain complexes come up in, they indicate a sort of “breaking up” of a space into smaller, more understandable pieces. In this post we take these basic ideas of chain complexes and abstractify them so that we can discuss them in general additive, and thus in abelian, categories.

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Chain Complexes

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Let $\mathscr{A}$ be an additive category. We will think about chain complexes in $\mathscr{A}$ as being objects in a certain full subcategory of the functor category $\mathscr{A}^\mathbb{Z}$, where $\mathbb{Z}$ is the category induced by the reverse usual ordering on $\mathbb{Z}$ (the fact that we are doing the reverse of the usual ordering on $\mathbb{Z}$ is for purely historical reasons–it makes no real difference whether we take the usual or reverse ordering). Ok, but getting a little more grounded, what exactly does this mean? Well, as we have noted countless times before an element of the functor category $\mathscr{A}^\mathbb{Z}$ is what we get when we take the Hasse diagram for $(\mathbb{Z},\leqslant_{\text{op}})$, which we will envision as

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$\cdots n+1\to n\to n-1\to\cdots$

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and at each integer place an object $X_n$ of $\mathscr{A}$ and each arrow place a morphism $\partial_n$ (we needn’t specify the morphism $X_{12}\to X_6$ since it is necessarily just the composition of the arrows $X_{12}\to X_{11}\to\cdots\to X_6$), so it would look something like

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$\cdots X_{n+1}\xrightarrow{\partial_{n+1}} X_n\xrightarrow{\partial_n}X_{n-1}\cdots$

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For reasons relating to the topological occurences of chain complexes the maps $\{\partial_n\}$ are called the boundary maps and often times the maps are denoted $\partial$, without the subscript, when no confusion will arise. We shall often times denote a chain complex as $(C_n,\partial_n)$ (where obviously this really denotes a set of objects and a set of maps) and may something like “let $\mathbf{C}=(C_n,\partial_n)$ be a chain complex). In fact, if I am feeling particularly lazy I may just say something to the effect of “let $\mathbf{C}$ and $\mathbf{D}$ be chain complexes” where what I really mean is that $\mathbf{C}=(C_n,\partial_n)$ and $\mathbf{D}=(D_n,d_n)$–it’s just so much more convenient to lay this notation out now and just write $\mathbf{C}$ and $\mathbf{D}$.

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Ok, so we have said that chain complexes shall be objects in a certain full subcategory of $\mathscr{A}^\mathbb{Z}$–but which subcategory? Well, the special property we want our elements of $\mathscr{A}^\mathbb{Z}$ to hold is that the composition of two adjacent maps is zero–in other words, $\partial_n\circ\partial_{n+1}=0$ for all $n\in\mathbb{Z}$. Of course, this implies that the maps $X_{n}\to X_m$ for any $m>n$ are necessarily going to be zero. Thus, we formally define any element of $\mathscr{A}^\mathbb{Z}$ satisfying this property to be a chain complex and let $\mathbf{Ch}(\mathscr{A})$ denote the  full subcategory of $\mathscr{A}^\mathbb{Z}$ consisting of chain complexes in $\mathscr{A}$.

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Recall that a morphism between $\mathbf{C}$ and $\mathbf{D}$ is tantamount to giving a set of morphisms $C_n\xrightarrow{f_n}D_n$ such that resulting diagram

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$\begin{array}{ccccccc}\cdots & C_{n+1} & \xrightarrow{\partial_n} & C_{n-1} & \xrightarrow{\partial_{n-1}} & C_{n-2} & \cdots\\ & {^{f_n}}\big\downarrow & & {^{f_{n-1}}}\big\downarrow & & ^{f_{n-2}}\big\downarrow & \\ \cdots & D_{n+1} & \xrightarrow{d_{n+1}} & D_n & \xrightarrow{d_n} & D_{n-1} & \cdots \end{array}$

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These are called, unsurpsingly chain maps–a chain map shall often times bet denoted $\{f_n\}$. Moreover, recall that the $\mathbf{Ab}$-category  structure on $\mathscr{A}$ is inherited by $\mathbf{Ch}(\mathscr{A})$ by taking two chain maps $\{f_n\}$ and $\{g_n\}$ and adding them “componentwise” to get $\{f_n+g_n\}$.

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There are a few special subcategories of $\mathbf{Ch}(\mathscr{A})$ that are worth mentioning. Namely, the category $\mathbf{Ch}_{\geqslant 0}(\mathscr{A})$ consisting of complexes $\mathbf{C}$ such that $C_n=0$ for $n<0$. The category $\mathbf{Ch}_b(\mathscr{A})$ consisting of those chain complexes which are bounded in the sense that there exists some $N$ such that $C_n=0$ whenever $N<|n|$.

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Our immediate goal right now is to discuss how $\mathbf{Ch}(\mathscr{A})$ is naturally an abelian category whenever $\mathscr{A}$ is, and along the way why $\mathbf{Ch}(\mathscr{A})$ is additive whenever $\mathscr{A}$ is. This will, of course, not only be useful in the sense that we get new abelian categories to play around with, but will force us to figure out what the important constructions (kernels, direct sums, etc.) are in our new category. So, let’s begin.

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The first thing that we’d like to do is pass from what we have now, an $\mathbf{Ab}$-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, it’s not hard to guess what the zero object is. Namely, let’s fix, once and for all, some zero object $0$ of $\mathscr{A}$. Then, we claim that the chain $\mathbf{0}$ consisting of $X_n=0$ and $\partial_n=0$ is a zero object for $\mathbf{Ch}(\mathscr{A})$. This is fairly obvious, right? Let $\mathbf{C}$ be any object of $\mathbf{Ch}(\mathscr{A})$. We know there exists unique arrows $0\xrightarrow{f_n}C_n$ and $C_n\xrightarrow{g_n}0$ and so clearly there are unique arrows $\mathbf{0}\to\mathbf{C}$ and $\mathbf{C}\to\mathbf{0}$ given by $\{f_n\}$ and $\{g_n\}$ respectively. Thus, $\mathbf{Ch}(\mathscr{A})$ is now preadditive having the zero object $\mathbf{0}$.

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References

[1] Weibel, Charles A. An Introduction to Homological Algebra. Cambridge [England: Cambridge UP, 1994. Print.

[2] Schapira, Pierre. “Categories and Homological Algebra.” Web. <http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf&gt;.

[3] Rotman, Joseph. An Introduction to Homological Algebra. Dordrecht: Springer, 2008. Print.