In this question I work in the category of chain complexes of Abelian groups concentrated in degree $\geq -1$ (Most of the complexes I am interested in are augmented)
There is a standard notion of tensor product of chain complexes. I have been studying a different product of chain complexes, a "join" which is identical to the usual tensor product except that the indices play a slightly different role: I define $(C\ast C')_n = \bigoplus_{i+j=n-1}C_{i+1}\otimes C'_{j+1}$
This has some interesting properties.
- This definition should have the effect that for abstract simplicial complexes $X,Y$, the chain complex associated to the join $X\ast Y$ is $C\ast C'$.
- I believe that if $\Delta^p$ and $\Delta^q$ are the standard simplices in $Ch(\mathbf{Ab})$, we should have $\Delta^p\ast \Delta^q \cong \Delta^{p+q-1}$, as in the category of topological spaces. (Here, all my complexes are augmented with $\mathbb{Z}$ in degree $-1$.)
- Under this monoidal product, $\Delta^0$ is a monoid. The induced monoidal functor $\Delta\to Ch(\mathbf{Ab})$ sending $[n]$ to the $n+1$-fold product $\Delta^0\ast\dots\ast \Delta^0$ is the standard embedding of the simplex category into the category of chain complexes sending $[n]$ to the $n$-simplex. We extend this to a functor $\mathbb{Z}[\Delta]\to Ch(\mathbf{Ab})$ from the category whose morphisms are formal linear combinations of maps of ordinals. The functor $\mathbf{SAb}\to Ch(\mathbf{Ab})$ of the Dold-Kan correspondence is the left Kan extension of this functor along the $\mathbf{Ab}$-enriched Yoneda embedding $\mathbb{Z}[\Delta]\to \mathbf{SAb}$.
- The category of augmented simplicial Abelian groups is made into a monoidal category by Day convolution, based on the standard monoidal product in the augmented simplex category. I believe that the Dold-Kan equivalence is actually strong monoidal with respect to these products (the Day convolution and the join of chain complexes respectively)
I am having a difficult time establishing any interesting relationship between the ordinary tensor product and the join. I believe that the join is very interesting. Obviously one can give a definition of the join in terms of the tensor product or conversely but I don't know of any natural justification for the shift in indices. I don't why a notion of delooping or suspension should be relevant here to establishing a connection between them. I don't know what the tensor product looks like when transported to the category of augmented simplicial Abelian groups, and how it relates to the Day convolution there.
Is there any known correspondence between these operations? We have two operations which are computed in an almost identical way except for a shift in indices; but with serious differences between them. The choice of indexing is not arbitrary as it must be chosen in a way that is compatible with the geometric meanings of the objects and the functors that connect the category of chain complexes to categories of simplicial complexes, simplicial Abelian groups, etc. For instance, the Eilenberg-Zilber theorem holds for the tensor product but it should not hold for the join.
