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I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper inverse/direct images functors $f_!\dashv f^!$, and derived tensor product/internal hom $(-\otimes) A\dashv \mathsf{hom}(A,-)$.

I thought the pairs $f^\ast\dashv f_\ast$ and the tensor-hom adjunction looked familiar from the category of sets, although in that case the inverse image functor is right adjoint to the direct image functor (see this question). After poking around on google I found out about "Grothendieck's six operations". In every place I looked though, these were discussed at a level far beyond mine and in contexts I do not know - quasi-coherent sheaves, spectra, étale cohomology...

What is the idea and significance behind Grothendieck's six operations in general, and in the context of sheaves (over topological spaces) in particular?

I don't mind basic concepts from enriched category theory if they help.

Arrow
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This is my understanding of this yoga. It may not be exactly what you seek and may differ from another person's point of view. Also I apologize for my bad english.

For Grothendieck, many things should have a relative version. So instead of considering just a space $X_0$, consider a morphism $f:X\rightarrow S$ thought as a family of spaces $s\mapsto X_s:=f^{-1}(s)$.

Here is an easy example of relative thinking : Say you want to attach numerical invariants to your first space $X_0$, for example its dimension, you should instead replace it by a function on $S$, and you get the relative dimension. Under some assumptions, this function behave quite nicely.

If now you want to attach invariants that are sets, or abelian groups, to your first space $X_0$. Well there is a very nice tool that do exactly what you want for the relative version : presheaves on $S$. And under some assumptions, these are closely related to sheaves.

The six operations arise this way if you want to study cohomology.

There are many cohomology theories out there, and many of them have their 6 operations. But they do not behave exactly of the same way, so let's say we work with ordinary cohomology of (nice) topological spaces.

It is also better to work with derived categories. Indeed, the cohomology of space is in fact a complex up to quasi-isomorphism, more than just the groups $H^i$. Also, for example, the functor $f^!$ exists only at the level of derived categories. If $X$ is a space, let $D(X)$ be the derived category of sheaves on $X$ and let $\mathbb{Z}_X$ be the constant sheaf with value $\mathbb{Z}$ on $X$.

Let start with the first functor $f_*$ or rather $Rf_*:D(X)\rightarrow D(S)$. Well, this is exactly the functor that computes cohomology. Indeed, if $f$ is proper, $(R^if_*\mathbb{Z}_X)_s=H^i(X_s,\mathbb{Z})$ (this is the so-called proper base change theorem). So this is exactly the relative version of cohomology. In fact, if $f$ is proper and smooth (=submersion), then $R^if_*\mathbb{Z}$ are local systems on $S$.

There is also $f_!$ or rather $Rf_!:D(X)\rightarrow D(S)$. It does the same thing, but with cohomology with compact support : $(R^if_!\mathbb{Z}_X)_s=H^i_c(X_s,\mathbb{Z})$. This times, one does not need to assume $f$ proper.

And of course the tensor product $\otimes$ gives the multiplicative structure on cohomology. One can then speak of derived analogue of the Kunneth formula...

Now, if $g:T\rightarrow S$, there is the functor $g^*:D(S)\rightarrow D(T)$. In some sense this is the one that justify the whole thing : this is the functor which correspond to changing the base from $S$ to $T$. For example, if $j:U\rightarrow S$ is the inclusion of an open subset, one can form $X_U=f^{-1}(U)$ and $j^*Rf_*\mathbb{Z}_X=R(f_{|X_U})_*\mathbb{Z}_{X_U}$. This is a (very easy) special case of a very deep one : the smooth base-change theorem. Changing bases is VERY useful. For example, base change to the universal cover, so that local systems become constant. Or in algebraic geometry, base change to the algebraic closure. And of course, taking stalks are already special cases of base change...

And finally, the functor $f^!:D(S)\rightarrow D(X)$ and the internal Hom are there to deal with duality. (Note that $f^!$ is right adjoint to $f_!$, not left). Instead of a global duality between $H^i_c$ and $H_{d-i}$, we now have local versions, allowing local computations and so on... Just for completeness, if $S$ is a point, $f_*f^!\mathbb{Z}$ is the Borel-Moore homology and $f_!f^!\mathbb{Z}$ is the ordinary homology (at least if $X$ is nice enough).

Roland
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    Could you please give a reference source to all this? – magma Jul 08 '15 at 22:33
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    I'm afraid the only references I can give you will just explain the construction and the properties of the six functors. I mentioned proper and smooth base change, but there are many more : projection formula, Verdier duality, gluing... Many books (eg Kashiwara, Schapira) give an exhaustive list of these properties. As for this "yoga", it came to me from practice, that is why it is really a personal point of view... – Roland Jul 09 '15 at 07:23
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    @Roland How is cohomology of a space in fact a complex up to quasi-isomorphism? What are the differentials? –  Sep 18 '15 at 10:25
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    @Exterior I meant, better to consider the groups $H^i$, we should consider the complex $C^\bullet(X)$ that computes it. However, the complex depends on the construction of the cohomology, (singular cochains, de Rham complex, global sections of an injective resolution of the constant sheaf, simplicial resolution...) but all those complexes are quasi-isomorphic, so give rise to single object (up to isomorphism) in the derived category. This is this object that is the most meaningful to study the space. – Roland Sep 18 '15 at 11:23
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    But seriously, though, what is a good reference for the six functor properties? For example, which Kashiwara and Schapira do you mean? (They do have 33 joint publications...) – Ben Webster Oct 01 '15 at 18:57
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    @BenWebster I meant there big book "Sheaves on manifold". You will have all the adjunction properties you can ask for... – Roland Oct 01 '15 at 18:59
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    @Roland : Thanks a lot for this answer this is really useful for me ! –  Nov 01 '15 at 18:56
  • @N.H. Glad it helps ! – Roland Nov 01 '15 at 18:57
  • @Roland is it possible to contact you by mail? – Arrow Nov 08 '15 at 21:15
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    @Arrow Sure, thought I do not want to give my email to everyone, how can I give it to you ? – Roland Nov 08 '15 at 21:23
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    There are these notes, probably Grothendieck himself. http://matematicas.unex.es/~navarro/res/sixoperations.pdf – ACL Nov 05 '16 at 22:49
  • Dear @Roland, thank you again for this wonderful answer. By "smooth base-change theorem" in the context of topology, do you mean Proposition 3.1.9 of Kashiwara-Schapira? – Arrow Dec 07 '18 at 23:49
  • Well, proposition 3.1.9 is just the adjoint of the proper base change theorem, thought sometimes it is indeed called the smooth base change theorem. I had in mind an isomorphism $g^*Rf_*\simeq Rf_*g^*$ when $g$ is smooth. This follows from proposition 3.1.9 and an identification of $f^!$ with a twist of $f^*$. – Roland Dec 08 '18 at 09:59
  • Dear @Roland, I don't know what a "twist" is. Is there a reference for the smooth base change theorem in topology? (e.g along a submersion) – Arrow Dec 09 '18 at 23:19
  • @Arrow A twist is simply a tensor product with an invertible object. For example a shift $[1]$, a Tate twist in algebraic geometry $(1)$ or the orientation twist $\mathfrak{or}$ in differential geometry. All of this is done in Kashiwara-Shapira. Proposition 3.1.9 indeed but also with 3.3.2 (see the discussion which follows on page 153). – Roland Dec 10 '18 at 20:11
  • Dear @Roland, where can I find applications (in the topological/smooth categories) of "base change to the universal cover"? Where does one find these beautiful things... – Arrow Dec 28 '18 at 18:09
  • @Arrow The base change to the universal cover is indeed useful in the study of vanishing cycles. There is a lot a literature on the subject, in algebraic geometry and in topology (other keywords on the subject : nearby cycles, perverse sheaves (another place where we have a 6 functors formalism btw), Picard-Lefschetz theory...). – Roland Dec 28 '18 at 20:52
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I don't have the time to expand the answer in a detailed but succinct account of those, but if you're more on the category theory side of the story I can throw some references which contain some information: it should appear evident why they are kind of related to the question

  1. J-B. Calmès and J. Hornbostel, "Tensor-triangulated categories and dualities", Theory and Applications of Categories, Vol. 22, 2009, No. 6, pp 136-198.

  2. Ryan Reich, "Obvious natural morphisms of sheaves are unique", Theory and Applications of Categories, Vol. 29, 2014, No. 4, pp 48-99.

  3. F. Hörmann, "Fibered Multiderivators and (co)homological descent" that contains all the prerequisites for

  4. F. Hörmann, "Six Functor Formalisms and Fibered Multiderivators"

fosco
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  • I've seen the papers by Hörmann before but felt they're above my head. Maybe I'll understand a little more of them now. Thanks! – Arrow Jan 04 '17 at 21:43