Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 11 of 11
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.
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”?
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.
@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.
@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 $C$.
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.
1 to 11 of 11