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Recently I’ve been looking a lot at the derived category of coherent sheaves on a scheme (specifically of Calabi-Yau threefolds). I’ve been told that it is “better” to consider this as a dg-category for whatever reason. I’m quite interested in exploring this, since I have no idea why this would be. For instance, the great book Fourier-Mukai Transforms in Algebraic Geometry by Huybrechts develops tons of theory and has lots of great beautiful theorems, but the phrase “dg-category” never once appears in the whole book. Do you really get much more from this point of view?
Anyway, I’m interested in writing notes here as I try to figure out what is going on here. I’m trying to take inventory of what pages already exist on this topic. I’ve found derived algebraic geometry, dg-category, derived category. I can’t find anything more on topic than those, so am I right in assuming that there seems to be very little about the derived category of sheaves or am I missing it somewhere?
This is an example of a general phenomenon:
a derived category is always the homotopy category of a stable (infinity,1)-category.
Generally, a homotopy category – by definition – is good for remembering equivalence classes of objects (as in equivalence in an (infinity,1)-category). But for nothing else!
Meaning: no actual category theory works in the homotopy category as expected. Meaning: no universal constructions, no limits, no colimits, etc.
The reason is that the correct universal constructions must be universal with respect to the higher degrees of freedom of the higher category (must be homotopy limits etc.) which is precisely the information that is discarded when passing to the homotopy category.
Now there are plenty of models and presentations for $(\infty,1)$-categories in general and stable (infinity,1)-categories in particular (see that entry for links, scroll down a bit). The main one being by linear A-infinity categories. These can always be strictified to dg-categories. Therefore many texts will tell you to use these.
Thanks. This is great to get me started. I guess one thing I’m really curious about right from the start is the following: If $X$ and $Y$ are two varieties (maybe say CY 3-folds), then there seems to be two notions going around that may or may not be equivalent. If $D^b(X)$ is equivalent as a triangulated category to $D^b(Y)$, is it still possible that they are not equivalent as A-infinity categories … or dg-categories?
The essence of this question being, the way I think about these things right now is in some sense the “decategorified” version, so if I care about these up to equivalence, do I actually lose something? Another example is that there is a construction due to Caldararu to show that it is possible for X, Y to be non-birational CY 3-folds and still have equivalent derived categories as triangulated categories. It would be interesting to know if using equivalence as A-infinity categories rules this possibility out.
If $D^b(X)$ is equivalent as a triangulated category to $D^b(Y)$, is it still possible that they are not equivalent as A-infinity categories … or dg-categories?
It is certainly in general so that two homotopy categories $Ho(\mathcal{C})$ ad $Ho(\mathcal{D})$ being equivalent says very little about whether the $\infty$-categories $\mathcal{C}$ and $\mathcal{D}$ are equivalent. In your case you are looking at homotopy categories of the special kind $D^b(X) = Ho(QC(X))$ etc. So for general $\mathcal{D}$ an equivalence $D^b(X) \simeq Ho(QC(X)) \simeq Ho(\mathcal{D})$ says nothing about an equivalence $QC(X) \simeq \mathcal{D}$.
Now of course if $\mathcal{D}$ itself is constrained to be of the form $QC(Y)$ maybe one can say more. I am not expert enough on that situation to know. But Zoran knows examples and counter-examples for such things. He can tell once he comes online.
The essence of this question being, the way I think about these things right now is in some sense the “decategorified” version,
That’s exactly what it is. $D^b(X)$ is the decategorification from $(\infty,1)$ to $(1,1)$ of $QC(X)$.
so if I care about these up to equivalence, do I actually lose something?
As I said, $D^b(X)$ knows everything about the equivalence classes in $QC(X)$ (the isomorphism classes in $D^b(X)$ are the equivalence class in $QC(X)$). So for that purpose $D^b(X)$ gives you everything you might want. A problem arises as soon as you want to look at universal construction of quasicoherent sheaves. $D^b(X)$ has forgotten essentially everything about these.
Another example is that there is a construction due to Caldararu to show that it is possible for X, Y to be non-birational CY 3-folds and still have equivalent derived categories as triangulated categories. It would be interesting to know if using equivalence as A-infinity categories rules this possibility out
Right. Again, I think, Zoran might know. I guess this is in the literature. But I don’t know off the top of my head.
Derived categories are sometimes not explicitly advertized as being homotopy categories (namely of the model category or $\infty$-category of complexes) but that’s precisely what they are.
However, more recently some people have started to use the word “derived” as a kind of synonym for “in higher category theory”. Which is a) very unfortunate while b) nevertheless almost aready fully standard and hence c) probably the source of your question.
Generally, a homotopy category … is good for remembering equivalence classes of objects…. But for nothing else!
Meaning: no actual category theory works in the homotopy category as expected. Meaning: no universal constructions, no limits, no colimits, etc.
I think that’s a bit of an exaggeration. For instance, products and coproducts work fine in a homotopy category, at least in the sense that if the $(\infty,1)$-category has them, then they are preserved by passage to the homotopy category (though the converse might not be true). And especially if you remember a bit more structure on the homotopy category, like making it a triangulated category, then you can do a fair bit more with it than just remembering equivalence classes of objects. A derivator is a way of adding sufficiently much structure to a homotopy category that you can do basically all the category theory you want to with it (though that “sufficiently much” is a lot of structure, but well-organized structure).
I agree with the general point, of course, that homotopy categories are not well-behaved categorically and usually (more usually than is done in practice, at least historically) one should use $(\infty,1)$-categories or at least derivators. I’m just saying they’re not as worthless as all that.
Sorry, I just realized a major source of confusion. The chain complexes of quasi-coherent sheaves, $QC(X)$, is a stable (infinity, 1)-category and when you take $Ho(QC(X))$ you get $D^b(X)$. I was confused because I thought somehow $D^b(X)$ itself was a stable (infinity, 1)-category and we were trying to take $Ho(D^b(X))$. Now this makes some more sense.
I’m not entirely sure that what I just wrote is correct. For instance, at Calabi-Yau category under examples, it says that $D^b(X)$ is an A-infinity category. Thanks.
For instance, products and coproducts work fine
Okay, sure.
A derivator is a way of adding sufficiently much structure to a homotopy category that you can do basically all the category theory you want to with it
Sure that adds in all the homotopy categories of all diagram categories. Which is then of course sufficient to compute homotopy limits.
I’m not entirely sure that what I just wrote is correct. For instance, at Calabi-Yau category under examples, it says that $D^b(X)$ is an A-infinity category.
Hm, I think we should edit that and at least clarify the terminology. I don’t think that’s particularly good notation. What is happening here is that people started to “fix” the deficiencies of the derived category for instance by passing from just a triangulated category to what is called an “enhanced triangulated category”. But that really amounts to making the full $(\infty,1)$-category behind it manifest.
If I would rewrite history I would ban the unspecific “derived” throughout. One needs to be careful in these discussions what kind of object exactly one is looking it.
Sure that adds in all the homotopy categories of all diagram categories. Which is then of course sufficient to compute homotopy limits.
And to do some other things, which on the surface might not look exactly like computing homotopy limits. For instance, I believe you can compute the mapping space between two objects (as an object of the homotopy category of spaces).
The entry is about the derived category of coherent sheaves on a smooth projective variety. In that c as, THERE IS NO LOSS OF INFORMATION from the enhanced triangulated category to the triangulated category, i.e. $(infty,1)$ gives no new information, as proved by Lunts and Orlov in the reference states in the entry: each derived category of that kind has a unique dg-enhancement. It is easier to work with triangualted one and one can get intofurther structure with hard work. So $D^b$ notation in this case may mean that one sometimes wants to look with it its canonical dg-enhancement; in mirror symmetry one often does the A-infinity enhancement without changing the notation. This uniqueness of extension is of large “philosophical” importance in physics where mainly such triangulated categories appear in practice.
I should also point out that the words “derived category” can abbreaviate “derived dg-category” and there are still he usual cases like bounded, unbounded, positive complexes…so the notation can be, depending on context, used in any of the cases. There is a difference in the case when the enhacement is not unique, and no important difference in geometric case.
Derived categories are sometimes not explicitly advertized as being homotopy categories (namely of the model category or ∞-category of complexes) but that’s precisely what they are.
However, more recently some people have started to use the word “derived” as a kind of synonym for “in higher category theory”. Which is a) very unfortunate while b) nevertheless almost aready fully standard and hence c) probably the source of your question.
Urs, derived category of an abelian category is, as you know and point out in model language, not its homotopy category, but the homotopy category of the category of the chain complexes in it. The idea is that in homological algebra one works with chain complexes modulo some relation of equivalence, which is obtained via a localization at acyclic complexes. A better version if to quotient a dg category of complexes by dg subcategory of acyclic complexes. This is called the derived dg-category or a dg-quotient. It is the idea of QUOTIENTing which is essential here more than the word homotopy which is 1-categorical quotienting. From the very start Grothendieck was looking for best possible way to quotient. Localization worked better than the quotient by a relation of equivalence which would be 0-categorical. Lyubashenko went into defining and studying A-infinity quotient version. The derived idea is idea of taking complexes and then quotient by acyclic. So the stable versions like dg and A-infinity and quasicategory case are just cases of the original derived as a quotient philosophy. I am not sure that all cases of derived categories as quotients come indeed from model categories. Lurie treats in such a way only bounded derived category of complexes over a ring, more can be done, but I am not sure of various nonabelian nonbounded extensions of derived categories like in Rosenberg etc. The idea of derived category is the idea that the quotient by acyclic complexes has some additional structure (like distinguished triangles, octahedra and so on). Quotient is the eternal idea, the categorical level is just a technicality.
I would not say that geometers use the word derived as a synonym for higher category theory. Instead they use it to say that the quotients etc. have to be made in the sense of total derived functors hence leading to infinity-stacks. I see no shift in intuition or in usage from the point of view of the algebraic geometry/homological algebra community.
Jim mentioned Vezzosi’s What is a derived stack, so here is the link
Urs, derived category of an abelian category is, as you know and point out in model language, not its homotopy category
Well, that’s not the original meaning. The original literature says “derived category” for the localization at quasi-isomorphisms.
Did you read what I wrote above ? I did not say that it was not a localization, but emphasised that it was a localization of the category of complexes, not of the original abelian category and the localization of the category of complexes, was FROM THE BEGINNING just a technicality to how to quotient by the subcategory of acyclic complexes in a sensible way for the homological algebra. Such localizations in teh setup of abelian categories were called quotient category by Serre and others. From the very beginning there was lots of ideas in practice about how to introduce the objects of homological algebra as complexes modulo complicated “equivalence” which realizes the idea of zeroing the acyclic complexes in the most sensible way. from the early days of Cartan-Eilenberg it was clear that we can take various resolutions equivalently, so the idea of taking some equivalence among complexes in systematic way is one of the first ideas of homological algebra motivating Grothendieck-Verdier from the start. They looked through adjustements and possibilities (like getting the homotopy equivalence first to get around the nonexistence of localization) and getting the additional structure which make derived category a derived category (like triangles, octahedra etc.). The 1980s idea of enhancing via dg-categories had a central subidea that the quotienting with dg-enrichement is better than with categorical localization; though formal quotienting and not enriching by hand has been fully de veloped only several years later by Keller and Drinfeld. Read the introduction of Gelfand-Manin book from 1988 for the historical account on how these ideas of how to quotient in the best way was a problem among experts from early 1960s.
Did you read what I wrote above ?
It’s true that I stopped after this:
Urs, derived category of an abelian category is, as you know and point out in model language, not its homotopy category,
That sounded to me like you are saying that the derived category means a model category structure and not just its homotopy category.
But I guess that’s not what you mean. I guess we agree on what’s going on.
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