Abstract
If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [ S , S ] C . An idempotent e of this ring will split the homotopy category: [ X , Y ] C ≅ e [ X , Y ] C ⊕ ( 1 − e ) [ X , Y ] C . We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L e S C × L ( 1 − e ) S C and [ X , Y ] L e S C ≅ e [ X , Y ] C . This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.
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