Subject: Chain complexes and simplicial model categories
From: Llorenç Rubió
Date: Tue, 23 May 2006 11:51:23 +0200
Dear all,
It seems to be folklore that the category Ch(R) of chain complexes of
R-modules is not a simplicial model category. That is said for example
after definition 4.2.18 in Hovey's Model Categories.
1. What the tensor product X\otimes K of a chain complex X and a
simplicial set K should be?
2. And the cotensor X^K?
3. The reason of not being a simplicial model category is in SM7 or
elsewhere?
Finally, in Jardine's paper Presheaves of Chain Complexes is stated
(lemma 2.5) that the category Ch_+(R) of positively graded chain complexes
of R-modules has the structure of a monoidal proper closed simplicial
model category,
via the Dold-Kan correspondence, but the chain complex tensor product is
not the standard.
Again there is not specified the tensor and cotensor by a simplicial set.
Llorenç Rubió