Definition 2.5.1.1. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$, and let $n$ be an integer. We will say that $C_{\ast }$ is concentrated in degrees $\geq n$ if objects $C_{m} \in \operatorname{\mathcal{A}}$ are zero for $m < n$. Similarly, we say that $C_{\ast }$ is concentrated in degrees $\leq n$ if the objects $C_ m$ are zero for $m > n$. We let $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\geq n}$ denote the full subcategory of $\operatorname{Ch}(\operatorname{\mathcal{A}})$ spanned by those chain complexes which are concentrated in degrees $\geq n$, and $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\leq n}$ the full subcategory spanned by those chain complexes which are concentrated in degrees $\leq n$.
2.5.1 Generalities on Chain Complexes
In this section, we provide a brief review of some of the homological algebra which will be needed throughout §2.5.
Example 2.5.1.2. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C \in \operatorname{\mathcal{A}}$ be an object, and let $n$ be an integer. We will write $C[n]$ for the chain complex given by where each differential is the zero morphism. Note that a chain complex $M_{\ast }$ is isomorphic to $C[n]$ (for some object $C \in \operatorname{\mathcal{A}}$) if and only if it is concentrated both in degrees $\geq n$ and in degrees $\leq n$.
Notation 2.5.1.3 (Cycles and Boundaries). Let $\operatorname{\mathcal{A}}$ be an abelian category (Definition In the special case where $\operatorname{\mathcal{A}}= \operatorname{ Ab }$ is the category of abelian groups, we will refer to the elements of $C_{n}$ as $n$-chains of $C_{\ast }$, to the elements of $\mathrm{Z}_{n}(C)$ as $n$-cycles of $C_{\ast }$, and to the elements of $\mathrm{B}_{n}(C)$ as $n$-boundaries of $C_{\ast }$. 
) and let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$. For each integer $n$, we let $\mathrm{Z}_{n}(C)$ denote the kernel of the boundary operator $\partial : C_ n \rightarrow C_{n-1}$, and $\mathrm{B}_ n(C)$ the image of the boundary operator $\partial : C_{n+1} \rightarrow C_ n$. We regard $\mathrm{Z}_ n(C)$ and $\mathrm{B}_{n}(C)$ as subobjects of $C_ n$. Note that we have $\mathrm{B}_{n}(C) \subseteq \mathrm{Z}_{n}(C)$ (this is a reformulation of the identity $\partial ^2 = 0$).
Definition 2.5.1.4 (Homology). Let $\operatorname{\mathcal{A}}$ be an abelian category and let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$. For every integer $n$, we let $\mathrm{H}_{n}( C )$ denote the quotient $\mathrm{Z}_{n}(C) / \mathrm{B}_{n}(C)$. We will refer to $\mathrm{H}_{n}(C)$ as the $n$th homology of the chain complex $C_{\ast }$. We say that the chain complex $C_{\ast }$ is acyclic if the homology objects $\mathrm{H}_{n}(C)$ vanish for every integer $n$. If $\operatorname{\mathcal{A}}= \operatorname{ Ab }$ is the category of abelian groups and if $x \in \mathrm{Z}_ n(C)$ is an $n$-cycle of $C_{\ast }$, we let $[ x ]$ denote its image in the homology group $\mathrm{H}_{n}(C)$: we refer to $[x]$ as the homology class of $x$. We say that a pair of $n$-cycles $x,x' \in \mathrm{Z}_{n}(C)$ are homologous if $[ x ] = [ x' ]$: that is, if there exists an $(n+1)$-chain $y$ satisfying $x' = x + \partial (y)$.
Definition 2.5.1.5 (Quasi-Isomorphisms). Let $\operatorname{\mathcal{A}}$ be an abelian category, let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in $\operatorname{\mathcal{A}}$, and let $f: C_{\ast } \rightarrow D_{\ast }$ be a chain map. We say that $f$ is a quasi-isomorphism if, for every integer $n$, the induced map of homology objects $\mathrm{H}_{n}(C) \rightarrow \mathrm{H}_{n}(D)$ is an isomorphism.
Remark 2.5.1.6. Let $C_{\ast }$ be a chain complex with values in an abelian category $\operatorname{\mathcal{A}}$. In practice, the homology objects $\mathrm{H}_{\ast }(C)$ are often primary objects of interest, while the chain complex $C_{\ast }$ itself plays an ancillary role. The terminology of Definition 2.5.1.5 emphasizes this perspective: a chain map $f: C_{\ast } \rightarrow D_{\ast }$ which induces an isomorphism on homology should allow us to view the chain complexes $C_{\ast }$ and $D_{\ast }$ as “the same” for many purposes (this idea is the starting point for Verdier's theory of derived categories, which we will discuss in §).
Remark 2.5.1.7 (Two-out-of-Three). Let $\operatorname{\mathcal{A}}$ be an abelian category and suppose we are given a commutative diagram of chain complexes in which the rows are exact. If any two of the chain maps $f$, $f'$, and $f''$ are quasi-isomorphisms, then so is the third. This follows by comparing the long exact homology sequences associated to the upper and lower rows (see Construction ).
Proposition 2.5.1.8. Let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in an abelian category $\operatorname{\mathcal{A}}$, and let $f,f': C_{\ast } \rightarrow D_{\ast }$ be a pair of chain maps. If $f$ and $f'$ are chain homotopic, then they induce the same map from $\mathrm{H}_{n}(C)$ to $\mathrm{H}_{n}(D)$ for every integer $n$.
Proof. Let $h = \{ h_ m \} _{m \in \operatorname{\mathbf{Z}}}$ be a chain homotopy from $f$ to $f'$, so that $f'_{n} - f_ n = \partial _{D} \circ h_{n} + h_{n-1} \circ \partial _{C}$. It follows that, when restricted to the subobject $\mathrm{Z}_{n}(C) \subseteq C_ n$, the difference $f'_{n} -f_ n = \partial _{D} \circ h_ n$ factors through the subobject $\mathrm{B}_{n}(D) \subseteq \mathrm{Z}_{n}(D)$, so the induced maps $\mathrm{H}_{n}(f), \mathrm{H}_{n}(f'): \mathrm{H}_{n}(C) \rightarrow \mathrm{H}_{n}(D)$ are the same. $\square$
Corollary 2.5.1.9. Let $f: C_{\ast } \rightarrow D_{\ast }$ be a chain map between chain complexes with values in an abelian category $\operatorname{\mathcal{A}}$. If $f$ is a chain homotopy equivalence, then it is a quasi-isomorphism.
For later use, we record the following elementary fact:
Proposition 2.5.1.10. Let $P_{\ast }$ be a chain complex taking values in an abelian category $\operatorname{\mathcal{A}}$. Assume that $P_{\ast }$ is acyclic, concentrated in degrees $\geq 0$, and that each $P_{n}$ is a projective object of $\operatorname{\mathcal{A}}$. Then $P_{\ast }$ is a projective object of the category $\operatorname{Ch}(\operatorname{\mathcal{A}})$. In other words, every epimorphism of chain complexes $f: M_{\ast } \twoheadrightarrow P_{\ast }$ admits a section.
Proof. Our assumption that $P_{\ast }$ is acyclic guarantees that for every integer $n \geq 0$, we have a short exact sequence
It follows by induction on $n$ that each of these exact sequences splits and that each $\mathrm{Z}_ n(P)$ is also a projective object of $\operatorname{\mathcal{A}}$. We can therefore choose a direct sum decomposition $P_{n} \simeq \mathrm{Z}_ n(P) \oplus Q_ n$, where the differential on $P_{\ast }$ restricts to isomorphisms $\partial : Q_{n} \simeq \mathrm{Z}_{n-1}(P)$. Since each $Q_{n}$ is projective and $f$ is an epimorphism in each degree, we can choose maps $u_{n}: Q_{n} \rightarrow M_ n$ for which the composition $f_ n \circ u_ n$ equal to the identity on $Q_{n}$. The maps $u_ n$ then extend uniquely to a map of chain complexes $s = \{ s_ n \} _{n \in \operatorname{\mathbf{Z}}}$, characterized by the requirement that each composition
is the sum of the maps $\partial u_{n+1}$ and $u_ n$. $\square$
We now specialize our attention to the category $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ of chain complexes of abelian groups, which we will endow with a monoidal structure.
Notation 2.5.1.11. Let $C_{\ast }$ and $D_{\ast }$ be graded abelian groups. We define a new graded abelian group $(C \boxtimes D)_{\ast } = C_{\ast } \boxtimes D_{\ast }$ by the formula Here the direct sum is taken over the set $\{ (n', n'') \in \operatorname{\mathbf{Z}}\times \operatorname{\mathbf{Z}}: n = n' + n'' \} $ of all decompositions of $n$ as a sum of two integers $n'$ and $n''$, and $C_{n'} \otimes D_{n''}$ denotes the tensor product of $C_{n'}$ with $D_{n''}$ (formed in the category of abelian groups). For every pair of elements $x \in C_{m}$ and $y \in D_{n}$, we let $x \boxtimes y$ denote the image of the pair $(x,y)$ under the canonical map
Proposition 2.5.1.12. Let $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ be chain complexes. Then there is a unique homomorphism of graded abelian groups satisfying the identity for $x \in C_{m}$ and $y \in D_{n}$. Moreover, this homomorphism satisfies $\partial ^2 = 0$, so we can regard the pair $( (C \boxtimes D)_{\ast }, \partial )$ as a chain complex.
Proof. For every pair of integers $m,n \in \operatorname{\mathbf{Z}}$, the construction
determines a bilinear map $C_ m \times D_{n} \rightarrow (C \boxtimes D)_{m+n-1}$. Invoking the universal property of tensor products and direct sums, we deduce that there is a unique map $\partial : (C \boxtimes D)_{\ast } \rightarrow (C \boxtimes D)_{\ast -1}$ with the desired properties. The identity $\partial ^2 = 0$ follows from the calculation
Notation 2.5.1.13. In the situation of Proposition 2.5.1.12, we will refer to $( (C \boxtimes D)_{\ast }, \partial )$ as the tensor product of the chain complexes $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$.
Warning 2.5.1.14 (The Koszul Sign Rule). Let $(C_{\ast }, \partial )$ and $( D_{\ast }, \partial )$ be chain complexes. There is a unique isomorphism of graded abelian groups $\tau : C_{\ast } \boxtimes D_{\ast } \rightarrow D_{\ast } \boxtimes C_{\ast }$ satisfying $\tau (x \boxtimes y) = y \boxtimes x$ for all $x \in C_{m}$, $y \in C_{n}$. Beware that $\tau $ is usually not a chain map: we have This can be remedied by modifying the isomorphism $\tau $: there is another isomorphism of graded abelian groups The isomorphism of $\sigma $ is a chain map (hence an isomorphism of chain complexes) by virtue of the calculation
Exercise 2.5.1.15 (Universal Property of the Tensor Product). Let $(C_{\ast }, \partial )$, $(D_{\ast }, \partial )$, and $(E_{\ast }, \partial )$ be chain complexes. We will say that a collection of bilinear maps satisfies the Leibniz rule if, for every pair of elements $x \in C_ m$ and $y \in D_{n}$, the identity holds in the abelian group $E_{m+n-1}$. Show that there is a canonical bijection from the collection of chain maps $f: C_{\ast } \boxtimes D_{\ast } \rightarrow E_{\ast }$ to the collection of systems of bilinear maps $\{ f_{m,n}: C_ m \times D_{n} \rightarrow E_{m+n} \} _{m,n \in \operatorname{\mathbf{Z}}}$ satisfying the Leibniz rule, given by the construction $f_{m,n}(x,y) = f( x \boxtimes y)$.
Remark 2.5.1.16 (Associativity Isomorphisms). Let $(C_{\ast }, \partial )$, $(D_{\ast }, \partial )$, and $(E_{\ast }, \partial )$ be chain complexes of abelian groups. Then there is a unique isomorphism of graded abelian groups satisfying the identity $\alpha ( x \boxtimes (y \boxtimes z) ) = (x \boxtimes y) \boxtimes z$. Moreover, $\alpha $ is an isomorphism of chain complexes: this follows from the observation that $\alpha ( \partial (x \boxtimes (y \boxtimes z) ) )$ and $\partial \alpha ( x \boxtimes (y \boxtimes z) )$ are both given by the sum for $x \in C_{m}$, $y \in D_{n}$, $z \in E_{p}$.
Construction 2.5.1.17 (The Monoidal Structure on Chain Complexes). Let $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the category of chain complexes of abelian groups (Definition 2.5.0.3). We define a monoidal structure on $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ as follows:
The tensor product functor $\boxtimes : \operatorname{Ch}(\operatorname{\mathbf{Z}}) \times \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$ carries each pair of chain complexes $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ to the tensor product chain complex $( C_{\ast } \boxtimes D_{\ast }, \partial )$ of Proposition 2.5.1.12, and carries a pair of chain maps $f: C_{\ast } \rightarrow C'_{\ast }$, $g: D_{\ast } \rightarrow D'_{\ast }$ to the tensor product map
For every triple of chain complexes $C = (C_{\ast }, \partial )$, $D = (D_{\ast }, \partial )$, and $E = (E_{\ast }, \partial )$, the associativity constraint
is the isomorphism of Remark 2.5.1.16.
The unit object of $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ is the chain complex $\operatorname{\mathbf{Z}}[0]$ of Example 2.5.1.2, and the unit constraint $\upsilon : \operatorname{\mathbf{Z}}[0] \boxtimes \operatorname{\mathbf{Z}}[0] \simeq \operatorname{\mathbf{Z}}[0]$ is the isomorphism classified by the bilinear map
Remark 2.5.1.18. Let $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ be chain complexes. The tensor product chain complex $(C_{\ast } \boxtimes D_{\ast }, \partial )$ of Proposition 2.5.1.12 is characterized up to (unique) isomorphism by the universal property of Exercise 2.5.1.15. However, the construction of this tensor product complex (and, by extension, the monoidal structure on $\operatorname{Ch}(\operatorname{\mathbf{Z}})$) depends on auxiliary choices. These choices are ultimately irrelevant in the sense that they do not change the isomorphism class of the monoidal category $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ or, equivalently, of the classifying simplicial set $B_{\bullet } \operatorname{Ch}(\operatorname{\mathbf{Z}})$ of Example 2.3.1.18. This simplicial set can be described concretely (without auxiliary choices): its $n$-simplices can be identified with systems of chain complexes $\{ C(j,i)_{\ast } \} _{0 \leq i < j \leq n}$ together with bilinear maps for $0 \leq i < j < k \leq n$ which satisfy the Leibniz rule $\partial (yz) = (\partial y)z + (-1)^{q} y(\partial z)$ together with the associative law $x(yz) = (xy)z$ for $x \in C(\ell ,k)_{r}$, $y \in C(k,j)_{q}$, $z \in C(j,i)_{p}$ with $0 \leq i < j < k < \ell \leq n$.