This might be a somewhat philosophical question in category theory. I sometimes have trouble understanding with some monoidal structures defined, why the one we choose are the "good ones". For example, why is the natural enrichment (or equivalently the tensor product) of chain complexes is defined this way. It seems to me like closed monoidal structure are often additional data, then how are the good definitions chosen ? Are there theorems ensuring that there are often only one closed monoidal structures on usual categories ? Or do we just define enrichment of new categories such that functors between new categories and old ones are (lax) monoidal ?
Choice of closed monoidal structure
category-theorymonoidal-categoriesphilosophy
Best Answer
Your last question gets at the point, I think. There are indeed multiple closed monoidal structures on chain complexes. Specifically, one has the pointwise structure, $(A\otimes B)_n=A_n\otimes B_n$, which is easily seen to be closed since colimits in chain complexes are levelwise. This gives a much simpler notion of monoid that the differential graded algebras which are the monoids for the usual monoidal closed structure on chain complexes.
However, it is differential graded algebras that arise naturally in examples, especially in topology. So, the usual monoidal structure is justified by the facts that (1) its monoids are interesting and (2) there is a very nice pair of functors (lax and oplax monoidal, strong monoidal up to homotopy equivalence), those of the Dold-Kan correspondence, between chain complexes and the simplicial abelian groups arising naturally from homology theory. For instance, this is already enough to see that the singular chain complex of a topological monoid is naturally a differential graded algebra, or that the chains on a simplicial Lie algebra are a differential graded Lie algebra, as used by Quillen in his approach to rational homotopy theory.