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In Residues and Duality, R. Hartshorne and A. Grothendieck say that there are a plethora of compatibilities that need to be shown in order to have a six functor formalism. For example, if $f:X\to Y$, $g:Y\to Z$, and $h:Z\to W$ are morphisms of ringed spaces, we have isomorphisms

and we would like to be sure that this diagram commutes. While it's simple to see that this diagram commutes, B. Conrad says in Grothendieck Duality and Base Change that some of these compatibilities are quite difficult to verify.

Finally, R. Hartshorne and A. Grothendieck says that "perhaps the language of fibred categories or the techniques of Giraud's thesis will supply what is needed".

I wonder what is the state of the art in this subject and, particularly, if fibred categories are any helpful.

P.S.: the book Triangulated Categories of Mixed Motives by D. C. Cisinski and F. Déglise uses this language to study six functor formalisms but, even though I tried, I admit that it is not clear to me if it solves this problem or not.

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If you have an abelian category $A$ with enough injective, for any functor defined on bounded below cochain complexes $\Phi\colon C^+(A)\to D$ sending cohain homotopy equivalences between degree-wise injective cochain complexes to isomorphism, there is a total right derived functor $R\Phi\colon D^+(A)\to D$. This means that there is a natural transformation $\eta_K\colon\Phi(K)\to R\Phi(K)$ for any bounded below cochain complex $K$ and that the pair $(R\Phi,\eta)$ is universal (initial) among all possible choices (i.e. it is the left Kan extension of the functor $\Phi$ along the localization functor from cohain complexes to the derived category). Furthermore, $R\Phi$ is not only a derived functor, but a universal one: this means that, for any functor $\Psi:D\to E$, the pair $(\Psi\circ R\Phi,\Psi(\eta))$ is the right derived functor of the composition $\Psi\circ\Phi$. All this works the same for unbounded complexes is $A$ is an abelian category and if we replace degree-wise injective cochain complexes by a suitable notion (e.g. fibrant of objects in a suitable Quillen model structure).

In particular, in practice, an identification of the form $R(g_*\circ f_*)\cong Rg_*\circ Rf_*$ never comes out of the blue, but rather always as derived functors: the isomorphism above is the unique map which is compatible with the canonical map $(g_*\circ f_*)(K)\to R(g_*\circ f_*)(K)$ and the image by $Rg_*$ of the canonical map $f_*(K)\to Rf_*(K)$ (functorially in $K$). Furthermore, even better, for any functor $\Psi$ whose domain is the codomain of $Rg_*$, the isomorphism $$\Psi(R(g_*\circ f_*)(K))\cong \Psi(Rg_*\circ Rf_*(K))$$ is the unique map which is compatible with the image by $\Psi$ of the canonical map $(g_*\circ f_*)(K)\to R(g_*\circ f_*)(K)$ and of the image by $Rg_*$ of the canonical map $f_*(K)\to Rf_*(K)$ (functorially in $K$).

For the compatibility problem you mention, we have to compare two maps from $R(h_* g_* f_*)$ to $Rh_* Rg_* Rf_*$. Using the universal property of $R(h_* g_* f_*)$, we just have to check that, for any cohain complex $K$, the two induced maps $$(h_* g_* f_*)(K)\to (Rh_* Rg_* Rf_*)(K)$$ agree, which comes right away from the compatibilities mentioned before.

As final remark: all this reasoning can be made in much more general context (e.g. with derived functors defined on homotopy categories of model categories). This extra level of generality means that all the reasoning takes place in a context where there are no possiblities to make sign mistakes because signs do not exist. All the ingredients of the proof are available since the late 1960's.

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  • $\begingroup$ Dear @Denis-Charles Cisinski, I'm not sure if what you said answers my question; I already knew that my diagram commutes. What I want to know is if there's some way to obtain all compatibility diagrams in six functor formalisms at once, without needing to do all of them one by one. $\endgroup$
    – Gabriel
    Sep 26, 2021 at 9:32
  • $\begingroup$ The kind of diagrams discussed above (non-derived vs derived functors) are not part of the six functor formalism: I consider this as obvious, and I would only consider derived functor so that such question would not even occur. Your question is not well defined, then: what would be "all compatibilities": being able to define precisely what that means is already chalenging. That said, the axiomatization of the six functor formalism is indeed a way to make this precise and to solve all coherence issues of interest. $\endgroup$ Sep 26, 2021 at 9:55
  • $\begingroup$ The ingredients do not go beyond cartesian fibrations as in Giraud's thesis, though. $\endgroup$ Sep 26, 2021 at 9:55
  • $\begingroup$ Finally, there are more coherence issues raised by higher category theory, if we consider the $(\infty,1)$-categorical version of derived categories. In the wok of Gaitsgory and Rozenlbyum, there is a description of the six functor formalism using the language of symmetric monoidal $(\infty.2)$-categories which certainly subsumes every other coherence issue from Residues and Duality. Gaitsgory and Rozenlbyum gave precise statements, and proving them in full is part of current research in higher category theory (already at a rather advanced stage). $\endgroup$ Sep 26, 2021 at 9:59
  • $\begingroup$ I don't understand what you mean by "I would only consider derived functors". Often, some functors on six functor formalisms are not derived functors (the $f^!$ in Verdier or Grothendieck duality, for example). Moreover, I indeed agree that defining properly what are those compatibilities is challenging. $\endgroup$
    – Gabriel
    Sep 26, 2021 at 10:19

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