The Stacks project

Lemma 22.4.2. Let $(A, d)$ be a differential graded algebra. The category $\text{Mod}_{(A, \text{d})}$ is abelian and has arbitrary limits and colimits.

Proof. Kernels and cokernels commute with taking underlying $A$-modules. Similarly for direct sums and colimits. In other words, these operations in $\text{Mod}_{(A, \text{d})}$ commute with the forgetful functor to the category of $A$-modules. This is not the case for products and limits. Namely, if $N_ i$, $i \in I$ is a family of differential graded $A$-modules, then the product $\prod N_ i$ in $\text{Mod}_{(A, \text{d})}$ is given by setting $(\prod N_ i)^ n = \prod N_ i^ n$ and $\prod N_ i = \bigoplus _ n (\prod N_ i)^ n$. Thus we see that the product does commute with the forgetful functor to the category of graded $A$-modules. A category with products and equalizers has limits, see Categories, Lemma 4.14.10. $\square$


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