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In a triangulated category with coproducts any idempotent splits.

Is there a proof of this fact different from that in Neeman, Prop. 1.6.8? In particular I'm looking for one which doesn't use the notion of homotopy colimit.

Thanks.

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    All proofs of this I know of rely on the same variant of the Eilenberg swindle. The triangulation is irrelevant, it's a fact about (pre-)additive categories with countable (co)products (or (co)powers). The argument is due to Freyd and can be found in his *Splitting homotopy idempotents,* in: Proceedings of the Conference on Categorical Algebra, La Jolla, CA, 1965, Springer, New York, 1966, pp. 173–176. However, the proof is not really different from that given in Neeman.2011-12-07
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    Here's an [MO thread](http://mathoverflow.net/questions/53006/how-do-i-split-a-homotopy-idempotent) on that argument.2011-12-07

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