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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 1st 2011
• (edited Jun 2nd 2011)

I have added reference pointers to Moritz Groth’s document on “Derivators, pointed derivators and stable derivators” to the relevant entries, such as stable derivator.

Mike, I forget if you mentioned that before or not. I only learned of his work today. Part of his PhD thesis with Schwede.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJun 2nd 2011

No, I wasn’t aware of that; it is very nice, thanks. I like that he uses “stable” instead of “triangulated”, that he doesn’t gratuitously turn around the direction of 2-cells, and his proof that combinatorial model categories give derivators is clean too. His construction of the “exceptional inverse image functors” is essentially the same as the one at pointed derivator, which is perhaps not surprising.

He does continue the odd tradition of saying “open immersion” and “closed immersion” instead of “cosieve” and “sieve”. I’ve never understood why anyone (Grothendieck?) chose to apply those words to functors, or what the intuition is for which is which.

I also don’t yet understand what the exceptional inverse image functors are used for. Groth shows how to construct them, but I don’t think he ever uses them.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeJun 2nd 2011
• (edited Jun 2nd 2011)

Mike, open immersion is not only a sieve: one requires that the functor is injective on objects, fully faithful AND that its image is a sieve. So I do not understand your complaint of why not calling it simply “sieve”. On the other hand, I am not sure if Grothendieck axiomatizes the properties of the morphism at the level of the corresponding direct image functor for the categories of quasicoherent sheaves; one should examine carefully to see. It is true that the direct image functor for open immersion is fully faithful, indeed, the inverse image functor is a localization; but now there is a slight trap here: one can view the localized category, but also one can look at the thick subcategory which one quotients at. That thick subcategory is reflective, but for a closed subscheme it is also coreflective. So it seems that we are in the right direction, but one should spend more time with this.

P.S. maybe one should in fact look at the properties of direct image functor among derived categories of qcoh sheaves instead (the properties are usually simpler there).

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJun 2nd 2011

According to sieve:

A sieve in a category C is a full subcategory closed under precomposition with morphisms in C.

In particular, that means that its inclusion functor is injective on objects and fully faithful, as for any full subcategory.

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