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    • CommentRowNumber1.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 29th 2011
    Having just discovered and read
    REMINISCENCES OF GROTHENDIECK AND HIS SCHOOL
    LUC ILLUSIE, WITH SPENCER BLOCH, VLADIMIR DRINFELD, ET AL.

    I was surprised to learn that derived cats date from 1964 and apparently prior to the cotangent complex
    and Quillen's homotopical algebra @ 1967??

    Does anyone know the history more accurately than that? and why the derived cat and
    homotopical algebra communities grew apart? Anyone maintain a foot in both camps?
    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeJan 29th 2011
    • (edited Jan 29th 2011)

    Jim, My understanding at the time was that Quillen’s HA was partially to extend the der. cat stuff to non-Abelian contexts. The der. cat was first in Verdier (état 0) which was to be his doctorat d’etat but was never really finished. At about that time there was also something by Puppe, (possibly later). AG did make comments in Pursuing Stacks about the then present state of derived category theory and linked it with Illusie’s thesis. The line he was pursuing was followed up by the Derivateurs work that he wrote later.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 29th 2011
    • (edited Jan 29th 2011)

    and why the derived cat and homotopical algebra communities grew apart?

    Is that really so?

    Is there a “derived category community” any more than there is a, say, “natural transformation community”?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 30th 2011

    Is there a “derived category community”

    Yes, I believe so. People like Amnon Neman, Daniel Murfet, people who work with derived and triangulated category approaches to alg geom and so on.

    • CommentRowNumber5.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 30th 2011
    I agree, though I meant especially linguistic communities. Equivalence does not mean equality!
    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 30th 2011
    • (edited Jan 31st 2011)

    @Urs: Yes, there definitely is. I would place Brian Conrad in the Derived Category community, while someone like Cisinski is a member of both communities. The nForum/nLab consists mainly of the Homotopy Category community, aside from Zoran, who I think is also a member of both.

    A derived categorist thinks of derived functors between abelian categories AA and BB as functors Ho(Ch A)Ho(Ch (B))Ho(Ch^-A)\to Ho(Ch^-(B)) which are left or right Kan extensions along the localization, while a homotopy categorist would probably say that the derived functor is a functor Ch ACh BCh^-A\to Ch^-B that restricts to a homotopical functor on a deformation retract of the homotopical category Ch ACh^-A, or more computationally, as a left Quillen functor between their corresponding model categories.

    That is, a homotopy-categorist preserves the distinction between functors and functors between homotopy categories.

    The derived category viewpoint has the advantage of not requiring a full model structure to talk about, although talking about homotopical rather than model categories rectifies this. A homotopical structure is the common generalization of model categories, categories of sheaves, and derived categories.

    • CommentRowNumber7.
    • CommentAuthorjim_stasheff
    • CommentTimeJan 31st 2011
    @Harry Gindi: The derived category viewpoint has the advantage of not requiring a full model structure to talk about, although talking about homotopical rather than model categories rectifices this. A homotopical structure is the common generalization of model categories, categories of sheaves, and derived categories.

    That's very helpful in the contemporary reality. As to history:Dear Prof Stasheff,

    I saw your question about the history of derived categories on the nlab, but I couldn't respond because I don't have an account. Here are a few data points that you might find helpful.

    1. In Recoltes et Semailles, Grothendieck writes:

    Vers l’année 1960 ou 1961 je propose à Verdier, comme travail de thèse possible, le développement de
    nouveaux fondements de l’algèbre homologique, basé sur le formalisme des catégories dérivées que j’avais
    dégagé et utilisé au cours des années précédentes pour les besoins d’un formalisme de dualité cohérente dans le
    contexte des schémas. Il était entendu que dans le programme que je lui proposais, il n’y avait pas de difficultés
    techniques sérieuses en perspective, mais surtout un travail conceptuel dont le point de départ était acquis, et
    qui demanderait probablement des développements considérables, de dimensions comparables à ceux du livre
    de fondements de Cartan-Eilenberg. Verdier accepte le sujet proposé. Son travail de fondements se poursuit
    de façon satisfaisante, se matérialisant en 1963 par un "Etat 0" sur les catégories dérivées et triangulées,
    multigraphié par les soins de l’ IHES. C’est un texte de 50 pages, reproduit en Appendice à SGA 4 1 en 1977 2
    (comme il est dit dans la note (63’ "))55(*).

    2. In the SGA 4 seminars in 63-64, they use derived categories all over the place.

    3. There is some more historical information in the preface of Hartshorne's Residues and Duality.

    In seems likely that Grothendieck knew the basic ideas by the late 50s and that Verdier had worked them out by the 62 or 63, although who knows when they became more widely known.

    Yours,

    James Borger



    James Borger writes:
    • CommentRowNumber8.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 31st 2011
    • (edited Jan 31st 2011)

    @Jim: Did Jim Borger include my comment in his email, or was that your response to it?

    Anyway, Dwyer-Hirschhorn-Kan-Smith’s Homotopy limit functors book gives a very useful perspective on model categories as a special case of homotopical categories. In particular, their main result is to define homotopy u-limits, u-colimits, Kan extensions, etc. all without the formalism of model categories.

    Given a functor between two small categories, u:ABu:A\to B and a homotopical category XX, a homotopy u-limit (resp. u-colimit) functor is defined to be a right (resp. left) approximation of the functor lim u:X AX Blim^u:X^A\to X^B (resp. colim u:X AX Bcolim^u:X^A\to X^B).

    Where a right (resp. left) approximation of a functor f:MNf:M\to N between homotopical categories MM and NN is defined to be a homotopically initial (resp. terminal) object of the homotopical category fι w(Fun w(M,N))f\downarrow \iota_w(Fun_w(M,N)) (resp. ι w(Fun w(M,N))f\iota_w(Fun_w(M,N))\downarrow f) where Fun w(M,N)Fun_w(M,N) is the homotopical subcategory of Fun(M,W)Fun(M,W) spanned by homotopical functors where the weak equivalences are the natural weak equivalences and ι w:Fun w(M,N)Fun(M,N)\iota_w:Fun_w(M,N)\hookrightarrow Fun(M,N) is the inclusion.

    Defining homotopically initial and terminal objects is kind of a pain directly (it is done in the book), but it is the same as saying that it is homotopy initial or terminal in the Dwyer-Kan simplicial localization. That is, an object xCx\in C for CC a homotopical category is homotopy initial (resp. homotopy terminal) if L HC(x,y)L^H C(x,y) (resp. L HC(y,x)L^H C(y,x)) is (weakly?) contractible for every object yy in CC.

    • CommentRowNumber9.
    • CommentAuthorjim_stasheff
    • CommentTimeFeb 2nd 2011
    @Harry: my posting responded to you and independently forwarded Borger's response to my initial posting.
    It's really the `ancient' history I'm trying to provoke.
    • CommentRowNumber10.
    • CommentAuthorMatanP
    • CommentTimeMar 27th 2012

    Without undermining the historical viewpoint, I ’m wondering if in our time there are substantial parts of derived category theory that are not captured by homotopical algebra. There is this issue of K-injectives/projectives in which I’m not sure there are known model structures that completely capture it. Maybe I’m just not updated.

    • CommentRowNumber11.
    • CommentAuthorjim_stasheff
    • CommentTimeMar 28th 2012
    @10 in our time there are substantial parts of derived category theory that are not captured by homotopical algebra.
    There is this issue of K-injectives/projectives in which I'm not sure there are known model structures that completely capture it. Maybe I'm just not updated.

    Certainly homotopical algebra is enlightened by model structures but not dependent on it. Same on the derived cat side?

    1967 does seem to be the earliest paper in mathscinet with the phrase `cotangent complex'