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' ]]>

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.

]]>It's really the `ancient' history I'm trying to provoke. ]]>

@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, $u:A\to B$ and a homotopical category $X$, a homotopy u-limit (resp. u-colimit) functor is defined to be a right (resp. left) approximation of the functor $lim^u:X^A\to X^B$ (resp. $colim^u:X^A\to X^B$).

Where a right (resp. left) approximation of a functor $f:M\to N$ between homotopical categories $M$ and $N$ is defined to be a homotopically initial (resp. terminal) object of the homotopical category $f\downarrow \iota_w(Fun_w(M,N))$ (resp. $\iota_w(Fun_w(M,N))\downarrow f$) where $Fun_w(M,N)$ is the homotopical subcategory of $Fun(M,W)$ spanned by homotopical functors where the weak equivalences are the natural weak equivalences and $\iota_w:Fun_w(M,N)\hookrightarrow Fun(M,N)$ 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 $x\in C$ for $C$ a homotopical category is homotopy initial (resp. homotopy terminal) if $L^H C(x,y)$ (resp. $L^H C(y,x)$) is (weakly?) contractible for every object $y$ in $C$.

]]>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: ]]>

@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 $A$ and $B$ as functors $Ho(Ch^-A)\to Ho(Ch^-(B))$ which are left or right Kan extensions along the localization, while a homotopy categorist would probably say that the derived functor is a functor $Ch^-A\to Ch^-B$ that restricts to a homotopical functor on a deformation retract of the homotopical category $Ch^-A$, 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.

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.

]]>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”?

]]>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.

]]>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? ]]>