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Someone recently said "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions". I have a rough idea why this is true ("don't throw away those higher homotopies -- you might need them some time in the future"), but I would be interested to have a more informed/detailed understanding of why triangulated categories are so abominable.

So, my question: Where are some good places to read about this "Down with triangulated!" philosophy?

I'm interested in this because in my research I have come across some categories which, very surprisingly, have a (semi-)triangulated structure. Perhaps I should be looking for an underlying A-infinity structure.

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Out of curiousity what do you mean by "semi-triangulated"? You might find it worthwhile to look at [A. Beligiannis and I. Reiten: ''Homological and Homotopical Aspects of Torsion Theories''] if you are interested in generalizations of the notion of triangulation (and you haven't already looked at it). I don't know how pre-triangulated categories in their sense fits into the enhanced picture. If someone knows this and would like to share that would be great (but I don't want to hijack the question if this isn't related). –  Greg Stevenson Oct 17 '09 at 2:05
    
By "semi-triangulated" I meant, vaguely, that not all of the axioms of a triangulated category are satisfied. More specifically, not every morphism has a mapping cone. But when morphisms f, g and fg each have a mapping cone, the octahedral axioms is satisfied. Also, the subset of morphisms which have mapping cones generates the category. It's the structure you would have for a subcategory of a triangulated category which is not a sub-triangulated-category. –  Kevin Walker Oct 17 '09 at 3:56
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There are many interesting triangulated categories without dg-models, i.e. the stable homotopy category. There is a new paper by Stefan Schwede (arxiv.org/abs/1201.0899) addressing the question of how and to what extent triangulated categories with such "topological models" can fail to have "algebraic models" in the spirit of dg-categories. –  Karol Szumiło Feb 18 '12 at 9:29

8 Answers 8

up vote 13 down vote accepted

I don't really think that triangulated categories are abominable, but they certainly have their problems which are a result of having forgotten the higher homotopies. For instance, non-functoriality of mapping cones can be fixed via dg-enhancement.

Another problem has to do with localization for triangulated categories. The fact that one can put a model structure on a suitable category of dg-categories and take homotopy limits for instance is a very useful thing and allows one to talk more meaningfully about gluing and working locally. One can do this in the derived category but only in quite simple situations and it requires extra structure for it to work well (e.g. the structure of a rigid tensor category compatible with the triangulation).

At least one good place to read in more detail about these problems and how to fix them using dg-categories is To¨en's notes on DG-categories which can be found at http://www.math.univ-toulouse.fr/~toen/swisk.pdf

I haven't really said anything about the A-infinity point of view, but I don't know it so well yet - hopefully someone else can say something. But I do know that at least one of the same problems rears its head namely that of wanting to work locally. For instance in homological mirror symmetry one would like to glue Fukaya categories of complicated things together from those of easier things. To do this one would certainly need to use the A-infinity point of view before taking derived categories and if it works it should be because one can take the appropriate homotopy limit (does the model structure to do this actually exist by the way?).

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The analogy that Nadler and I like is that if (oo,1)-categories are like vector spaces, then model categories are like vector spaces with a fixed basis and homotopy categories are like vector spaces mod isomorphism - i.e., dimensions of vector spaces. Which of the three would you rather work with?

In more details I'll take the low road and quote my paper with Francis and Nadler:

The theory of triangulated categories is inadequate to handle many basic algebraic and geometric operations. Examples include the absence of a good theory of gluing or of descent, of functor categories, or of generators and relations. The essential problem is that passing to homotopy categories discards essential information (in particular, homotopy coherent structures, homotopy limits and homotopy colimits).

This information can be captured in many alternative ways, the most common of which is the theory of model categories. Model structures keep weakly equivalent objects distinct but retain the extra structure of resolutions which enables the formulation of homotopy coherence. This extra structure can be very useful for calculations but makes some functorial operations difficult. In particular, it can be hard to construct certain derived functors because the given resolutions are inadequate. There are also fundamental difficulties with the consideration of functor categories between model categories.

Anyway, in characteristic zero, the theories of dg categories, Aoo categories and stable oo,1 categories are all the same, and provide a "promised land" between the two extremes. (The problems with localization of the Fukaya category are very serious, but are issues in geometry - the existence of instanton corrections - and not in the theory of A_oo categories). Perhaps the most useful result in this context that doesn't have analogs for dg or Aoo categories is Jacob Lurie's oo-Barr-Beck theorem, which has lots and lots of consequences - such as descent theory or the generation result that Ben mentioned. Another is having a good theory of tensor products of categories and internal hom of categories, neither of which works in the two extreme theories.

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Oops -- in the last paragraph I meant to say "the most useful result .. that doesn't have an analog for" triangulated or model categories .."is Barr-Beck"... Jacob's result certainly applies to (pretriangulated) dg or Aoo things, since they're the same as char.0 stable (oo,1)-categories.. –  David Ben-Zvi Oct 18 '09 at 15:17

I know other people have longer answers, but I have a short one: a dg-category can be reconstructed by knowing the Ext-algebra of any generating object as a dg-algebra, in the same way that an abelian category can be reconstucted from the endomorphisms of a projective generator. This isn't true if you only remember the Ext-algebra as a graded algebra; you might have needed those higher homotopies.

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I have two questions concerning your post: The Ext-algebra you're talking about - is it the endomorphism dg-algebra or is there some construction involved? Could you post a reference for the construction of the equivalence you mentioned and give an example of where this fails if you take only the graded algebra? –  Garlef Wegart Feb 8 '10 at 18:45
    
Yes, it's the endomorphism dg-algebra (I tend to think of it as an Ext-algebra because I'm usually thinking of dg-enhancements of triangulated categories. The equivalence is Hom with the object. I read about this in the survey paper on Keller's website, but the theorem is easy once you have the courage to believe it. –  Ben Webster Feb 8 '10 at 19:35

I don't think that triangulated categories/homotopy categories should be struck down, but their use is mostly as a place to examine the consequences of your actions. They're very difficult places to build things. You can't talk about topos theory in any derived context because you require more coherences than the cocycle condition in order to patch things like chain complexes using weak equivalences. There are complicated obstructions to rectifying commutative diagrams in the derived category to diagrams that honestly come from the category itself, and more generally you can't reconstruct the derived category of maps, or diagrams, from the original derived category.

You can do this stuff with dg-categories, or categories enriched in spaces, or A-infinity things. The derived category, or homotopy category, or stable homotopy category, or whatever is appropriate is then a good place to compute results.

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The point is that all these concepts (triangulated cat, dg-cat, quasitriangulated dg-cat, A-oo cat) are really approximations and/or presentations of stable (oo,1)-categories . Every triangulated category that appears in nature is the homotopy category of a stable (oo,1)-category. Hence working just with triangulated categories is wrong in just the same way as just working with derived categories is wrong: because all higher homotopies have been divided out all higer universal constructions such as homotopy limits and colimits have been lost.

See nLab:stable (oo,1)-category and references given there. See in particular the section on "alternative models".

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Concerning Greg Stevenson's question on how the A-oo point of view relates:

Recall that a dg-category is a category strictly enriched in chain complexes. An A-oo-category is a category weakly enriched in chain complexes.

But every A-oo-category is A-oo-equivalent to a dg-category. So dg-categoies are the "semi-strictification" of A-oo categories.

As such the relation between A-oo categories to dg-categories is analogous to that of quasi-categories to simplicially enriched categories: the former in each pair is the "more generic model", the latter the "more strictified model", but both are equivalent. The first pair models stable (oo,1)-categories, the second general (oo,1)-categories.

A great reference for this Bespalov et al's text "Pretriangulated A-oo categories". This is among the references that can be found at nLab:A-oo category.

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Thanks Urs, although I was actually asking about the notion of pre-triangulated category in the sense that suspension is not necessarily invertible but one has compatible left and right triangulated structures. This class of categories then includes triangulated categories, abelian categories (and quasi-abelian categories if memory serves) as well as additive closed model categories (well at least their homotopy categories). These categories turn out to be a good place to do torsion theory. I was wondering if they had a homotopy coherent analogue? –  Greg Stevenson Oct 17 '09 at 12:03
    
Although that is a very helpful response - I had forgotten that I did also ask about it in my actual answer and I think this makes it very clear what is going on. –  Greg Stevenson Oct 17 '09 at 12:12
    
It might be I still don't exactly see what precisely the question is, but just for the record: more on the pre-triangulated dg-story is here: ncatlab.org/nlab/show/pretriangulated+dg-category –  Urs Schreiber Oct 17 '09 at 15:03

One nice thing about the (∞,1)-categorical point of view is that being a stable (∞,1)-category is a property of an (∞,1)-category, and not extra structure. So whatever you have, you can probably find an (∞,1)-category version of it, and look for what properties it satisfies--but you probably don't need to look for extra structure.

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Hm, maybe your question really is what is the problem with passing to homotopy categories?

So there are entities that just happen to be (oo,1)-categories. Passing to their homotopy catgeory (and derived categories are a special case of that) means decategorifying them to a 1-category. This does retain all the information of equivalence classes of objects, which may be all one cares about in a given situation, but it loses most of the structural relation between these objects. Universal constructions (limit, colimit, adjoints, extensions) in the homotopy category don't produce the correct results that these universal constructions produce in the original (oo,1)-category.

For more on this see nLab:homotopy category of an (oo,1)-category

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