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edited model structure on sSet-presheaves a little
Hey Urs, do you know why they (Toen-Vezzosi) specifically used that model structure? It seems like it’s a pretty annoying one to describe, and some other pages here show that it’s Quillen equivalent to a bunch of much simpler ones.
Hi Harry,
i am not sure what you mean, could you expand? This is just the standard projective model structure on simplicial presheaves, set up in the enriched context.
Yes, but there are like 30 model structures on simplicial presheaves. This one (the local projective one) seems awfully complicated.
I am still not sure what you mean. The porjective and injective model structures on simplicial presheaves are those that are easy to state. The intermediate model structures are more complicated. But they are almost never used (Jardine in his lectures points out a single article that uses an intermediate model structure).
@Urs: There is a global model on simplicial presheaves defined by objectwise fibrations and weak equivalences. This is the “global projective model structure”. All of Toen’s work (including his lecture notes on DAG and simplicial commutative rings, etc) use the local projective model structure (the one that also encodes descent and hypercovers and stuff).
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