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
\[ (C \boxtimes D)_{n} = \bigoplus _{n = n' + n''} C_{n'} \otimes D_{n''}. \]
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
\[ C_{m} \times D_{n} \rightarrow C_ m \otimes D_{n} \hookrightarrow (C \boxtimes D)_{m+n}. \]