UHHH I COULDVE STOOD TO HEAR THIS EARLIER
Conversation
What I want for Christmas is... a conceptual proof of the Dold-Kan equivalence (and of its lax/oplax monoidal structures).
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are you familiar with the proof that proceeds by just calculating explicitly the cauchy completion of the free Ab-enriched category on the simplex category? (cauchy completion meaning adjoin finite biproducts, then split idempotents)
(i am not familiar with it)
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Where is this proof written? I would like to become familiar...
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i don't know where i saw this (there has to be a proof like this by general facts about morita equivalence) but it looks like this paper by lack and street does something like it:
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>there has to be a proof like this by general facts about morita equivalence
...does there?
feel like im missing some chunk of intuitions here...
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if A, B are two small Ab-enriched categories, TFAE: 1) [A^{op}, Ab] \cong [B^{op}, Ab], 2) [A^{op}, C] \cong [B^{op}, C] for any Ab-enriched category C which is cauchy complete (has finite biproducts and split idempotents), 3) the cauchy completions of A and B are equivalent
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the argument is basically that the cauchy completion of A is the subcategory of compact projective objects in [A^{op}, Ab]. now take A to be the free Ab-enriched category on the simplex category and B to be the free chain complex