Something about a similar setup in algebraic topology is here
Possibly “Hartshorne: Residues and duality” could be a good reference
The six functors formalism give rise to (“all”?) spectral sequences… ???
In Voevodsky (and Deligne): Lectures on cross functors, there is a discussion of the formalism of the four Grothendieck operations. This is maybe superseded by Ayoub’s work. However, the lectures have some nice brief background info on the formalism (for etale sheaves), including PD and the four kind of (co)homology, indicating where there are difficulties. There is not really any material on motivic homotopy theory, it just says that such applications will be given “later”.
Outline: 4 chapters, of which the first 3 are in his thesis.
2 types of applications: (A) Prove properties of general motives by “devissage” using the six operations, for example by reducing to the case of Tate motives. Schur-finiteness application. (B) Consctruction of motives and motivic cohomology classes. E.g. appl to polylogs. Remark: Applications restricted since motives over fields are not well understood, although we can reduce the study of motives over general schemes to motives over fields. In particular no motivic t-structure.
CHAPTER 1:
2-categorical preliminaries, cross functors. Abstact notion of “exchange structure” for a couple of 2-functors from a cat to a 2-cat. Extension of 2-functors. Lots of technical stuff. The end product of section 6 appears to be a construction of a cross-functor on the cat of quasi-projective S-schemes. A cross functor is four 2-functors together with four exchange structures, satisfying certain axioms. Section 7 gives for any quasi-projective S-morphism , a 2-morphism , which is a 2-isomorphism if is projective. In particular, get projective base change. This stuff should simplify the proofs of fundamental theorems in SGA4, for the derived categories of etale sheaves of L-modules, where L is a finite ring.
Section 4 contains the def of stable homotopical 2-functor, from (Sch/S) to TR (cat of triangulated cats). Six axioms: Empty scheme, right adjoint, left adjoint (for smooth f), localisation, homotopy invariance, stability. The main theorem now says essentially that a stable homotopical 2-functor , also denoted , can be extended, essentially uniquely, to a cross functor. I am guessing here that all the four 2-functors in the cross functor assigns the same triangulated cat to a given scheme, but a morphism between schemes is sent to the four different functors which are normally thought of as the four operations. Details in section 1.4.1.
For the generality of the scheme S, and the def of Sch/S, see section 1.3.5.
CHAPTER 2:
Many theorems in etale cohomology are proven using devissage via the proper and smooth base change theorems. Examples: (1) Constructibility of the higher direct images of constructible sheaves under a finite type morphism. (2) Stuff on cohomological dimension of the higher direct image functor (i.e. Artin’s thm on affine morphisms). (3) The formalism of Verdier duality (which relies on the notion of constructible sheaf).
Want to make precise the notions of constructibility and cohomological dimension in the motivic setting as well as in the general setting of a stable homotopical 2-functor.
In order to use compacity as a notion of constructibility, it is necessary to have small sums in our triangulated cats. Therefore we use a different notion. It appears that this uses a class of objects fixed in advanced, example given in the case of SH. Similar discussion of cohomological dimension, relating to the notions of t-structure, some stuff about the homotopical t-structure on DM(k) and SH(k) for k a field.
Section 1: Background on triangulated categories. Monoidal categories, modules and projectors (aiming at the coherence problems in Verdier duality theory I think). 1.7: Discussion on various hypotheses on stable homotopical 2-functors. End of section 1: Brief intro to RoS for general schemes. Statement, known to be true for base schemes essentially of finite type over a field of char zero, conjecturally true for other schemes such as finite fields and DVRs of unequal char. Defs and basic theorems on alterations, apparently pre-Gabber.
Section 2: Analogues of the contstructibility and cohomological dimension theorems in the etale setting. For simplicity, assume S is the spectrum of a perfect field admitting RoS. Thm: For f quasi-proj, all four functors sends constructible objects to constructibles. For cohomological dimension question (i.e. t-exactness), the situation is more complex, some detiails given.
Section 3: Verdier duality. Coherence problems, surmounted. Def of two pairings, the first of which lies behind most of the standard formulas relating the 4 operations, tensor prod and internal hom. Existence and uniqueness of dualizing objects, and consequences.
Section 4: Def of stable and homotopical algebraic derivator. Extension of the 2-functor SH to an algebraic derivator. Vanishing cycles in the setting of a stable and homotopical algebraic derivator, I think DM and SH are special cases.
CHAPTER 3: Nearby cycles in the motivic setting.
Could work with the specific stable homotopical 2-functors SH and DM, but prefer to work with a general algebraic derivator. Several advantages including applications to etale and Hodge theory, and compatibility with realizations, and coherence problems.
Key notion: System of specialisation. The idea seems to be the following: We start with a diagram which looks like the inclusions of the generic and closed points of a DVR, but can be much more general, call the middle “total” scheme B. Let H1 and H2 be two stable homotopical 2-functors over the “generic point” and the “closed point” respectively. Then a system of spec. from H1 to H2 is a “compatible” family of functors from H1 of the generic fiber to H2 of the special fiber; this data should be given for each scheme over B. Examples: (1) The canonical system of spec. associated to a stable homotopical 2-functor on Sch/B. (2) Classical etale settting over a strictly henselian trait.
Much more, including monodromy stuff, omitted for now.
CHAPTER 4: Construction of stable homotopical 2-functors (in particular DM and SH I think)
OUTLINE:
Section 1: Model cats and stable model cats.
Section 2: Hirschorn’s localisation theorem. Notion of accessibility, gives a notion of size of objects in model cats. Small object argument, cellular complexes. Model cats presented by cofibrations.
Section 3: Stabilisation of model cats, following Hovey, wrt a fixed left Quillen endofunctor. Uses the notion of F-spectra (symmetric, nonsymmetric etc). We start by studying spectra in abstract categories. Functoriality and comparison questions. Specialisation to the case where the functor is the tensor product by a cofibrant object in a symmetric monoidal model category. Def of tensor prod of symmetric spectra, and proof that stable model structure is symmetric monoidal.
Section 4: top-local model str on the cat of presheaves on a site with values in a model cat of “coeffs” which is assumed stable. Criterion for when a coninuous functor of sites induces a Quillen adjunction.
Section 5: Construction of the algebraic derivator SH. Various details, including compact generation.
Further notes from section 3:
For any unital graded monoid Phi in the cat of groups (e.g. the standard sequence of symmetric groups), and any Phi-symmetric endofunctor F, we have a notion of (F, Phi)-spectrum in an arbitrary cat C. If C is a model cat M, have notion of levelwise WE, and also: a projective cofib (injective fib) is a map satisfying the relevant lifting property wrt levelwise trivial fibs (cofibs). Under some technical hyps, including that F has a right adjoint which is accessible, get two model structures the cat of (F, Phi)-spectra, both with levelwise WEs (Prop 4.3.21). The model cats here are all presentable by cofibrations. These model structures are called the projective unstable and the injective unstable model structure. I think projective cofibs are levelwise cofibs.
Construction of Quillen adjunctions relating cats of spectra for different choices of Phi and F.
Def: A certain class R of morphisms in M, p 239, going from the (p+1)-suspension of F(X) to the p-suspension of X, for X cofibrant. Here Sus_p (def on p 227) is the left adjoint to Ev_p (notation my own).
Def: Assume M left proper, presentable by cofibrations, and that G is accessible. Def of projective stable and injective stable model structures by Bousfield localisation at the class R.
Simple description of R-local objects, looks to me like Omega-spectra. Actual def of Omega-spectra, slightly different.
The fibrant objects in the projective stable structure are the levelwise fibrant Omega-spectra.
Quillen adjunction comparison map between the stable projective structures associated to two different (F, Phi) inputs.
The up and down shifts give a Quillen endo-adjunction. wrt the projective stable structure. More comparison maps for different input data.
If (F, G) is a Quillen eq, then get a Quillen equivalence between M and (F, Phi)-spectra with the projective model structure.
Def: Symmetric element of . A symmetric element induces a left Quillen endofunctor on (F, Phi)-spectra wrt the projective unstable and the projective stable model structures.
Thm: Assume the existence of a symmetric element. Consider the projective stable model str. Then the induced left Quillen endofunctor is in fact part of a Quillen equivalence. Also, the shift pair is a Quillen equivalence. Furthermore, these two Quillen equivalences are related by a certain natural transformation.
Another comparison theorem (4.3.40) for different input data, idea of relative symmetric element. Several substatements. Always projective stable structure it seems.
Prop 4.3.42 Comparison between levelwise projective and stable projective structures, equiv under some hyps.
Let’s say GSS for (F, Phi)-spectrum. Can consider GSS objects in a cat of GSSa, have notion of biprojective bistable model structure, and compatibility relations (commutativity). (Prop 4.3.52)
More stuff about bispectra, feeding back into the proof of 4.3.40.
Paragraphe 4.3.4.Work with nonsymmetric spectra in this paragraph. Usual hyps on M. Consider a certain technical hyp (hyp56), satisfied in all cases of interest to us.
Prop 4.3.57: A certain kind of endofunctor on the levelwise homotopy cat of nonsymm spectra preserves stable equivalences.
Prop on levelwise spectra and a certain functor being a stable equivalence.
Thm: On a simple Omega-spectrum model for the R-localisation of a levelwise fibrant spectrum.
Par 4.3.5: Spectra in (unital symmetric) monoidal model cats.
For T a cofibrant object, can take F to be tensoring with T. The symmetric group acts on the n-fold tensor prod of T, and therefore we have cats of symmetric and nonsymmetric spectra SSP and SP. Some technical assumptions. We will see that SSP is a symmetric monoidal model cat (if it is stable???) and that under certain hyps, SP and SSP are Quillen equivalent.
Given a fairly general symmetric monoidal cat C, can identify symmetric spectra as left modules in Suite(Sigma,C) over a certain unital commutative algebra. Exploitation of this idea to get monoidal structure (for unstable and stable model str, under some hyps). Suff crit for SSP to be stable: I think this T should be isomorphic to a suspension in Ho(M); this is clearly true if M itself is stable.
Thm 4.3.79: Suff condition for SP and SSP to be Quillen equivalent (1) Voevodsky’s 123-condition, (2) Some transfinite composition stuff, and (3) SSP is stable.
Further notes from section 4
Section 4: Model cats of sheafy nature
Goal: Explain Jardine’s result on model structure on the cat of sheaves on a site. Key difference: we require the target cat to be a “cat of coeffs”, in particular to be stable, so sSet is excluded.
First some presheaf theory, apparently saying that a presheaf cat is cartesian closed, under “obvious” hyps.
Def of site in terms of covering sieves, and def of sheaf. Some basic notions about sheaves.
Thm: If C is presentable, we have a sheafification functor, which commutes with finite limits.
Various properties of sheafification.
Sectino 4.4.2: Presheaves with values in a model cat.
Def of injective and projective model str (Objectwise WE, and a projective fibration is an objectwise fibration). The projective model str is cofibrantly generated if the original M was so. Same for injective str but looks much harder.
Lemma on when tensor-Hom pair gives a Quillen adjunction, nonobviuos statement, different for inj and proj.
Lemma 4.4.20 on a family of representable functors being conservative, possible useful (guessing wildly) for showing that a known functor agrees with a functor constructed by a representing object.
Cor: If the target M is stable then so is PreShv, with both model structures.
Now to Jardine’s top-local model structure, in a restricted setting. Def of homotopically compact object. Def of “cat of coeffs”, notably stable with homotopy category probably compactly generated. Examples: Spectra and symmetric spectra over simplicial sets, AND complexes of left R-modules, when R is any ring and the model structure is the projective one, i.e. fibrations are the surjections.
Def of -sheaf ass to two objects H and K in PreShv(S, M). Def of top-local morphism in PreShv(S,M). Technical proof of the fact that we can localise at the class of top-local morphisms. Get top-local projective and injective model structures. WEs are exactly the top-local equivalences. With hyps as above, the top-local model structures are left proper and stable.
“Top-local model structure pass to the category of sheaves”.
Prop: If a morphism of presheaves becomes an iso after sheafification, then it is a top-local equivalence.
Def of top-local model structure on the category of sheaves: WEs are top-local equivalences, fibrations (inj and proj) inherited from the presheaf category, cofibs by lifting property.
Elementary functoriality: Recall that a functor between two sites induces on presheaves by composition; it has a left adjoint . Here presheaves take values in an arbitrary complete and cocomplete cat. If the target cat is a model cat, we get a Quillen adjunction wrt the projective model structures. Also statements about morphisms of sites, and the sheaf categories. Notion of pseudo-morphism of sites. These induce Quillen adjunctions on the presheaf cats wrt the local projective model structures.
More stuff about adjunctions and about sheaves, omitted here.
Thm 4.4.60: Roughly: A pre-morphism of sites compatible with two P-structures (cf small and big sites) induces a Quillen adjunction on presheaves, wrt the projective local structures.
Monoidal stuff: under mild conditions, presheaves with values in a (symmetric) monoidal model cat inherit the structure of a (symmetric) monoidal model cat, true for various model structures.
Finally, construction of the 2-functor SH, and verification that it is a homotopical and stable algebraic derivator. Details omitted.
Just a remark on the object T, the “sphere”. Bottom of p 325: We fix a projectively cofibrant object T in PreShv(Sm/S, M) such that for every smooth S-scheme U, the endofunctor on M is accessible. Then towards the end (p 348) one maybe needs to assume that T is equivalent to , but I am not sure what the notation Cof means. For the actual construction of the (M a cat of coefficients), T should be a cofibrant presheaf with a WE to this Cof.
nLab page on Grothendieck's six operations