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I added to Quillen bifunctor as a further "application" the discussion of Bousfield-Kan type homotopy colimits.
At some point I want to collect the material on homotopy (co)limits currently scattered at Bousfield-Kan map at weighted limit and now at Quillen bifunctor into one coherent entry.
After pointing to it from MathOverflow here, I polished a tiny little bit some parts of homotopy limit.
More needs to be done here.
somehow I felt like expanding the section global definition at homotopy limit – even though I should really be doing something else.
I amplified the relation to homotopy Kan extension a bit more, spent more words on the special case of simplicial model categories, on the enriched Quillen adjunctions that limit and colimit form there, and then at the end wrote some paragraphs on the derived weigheted colimits and how they can be used to computed unweighted derived colimits.
This duplicates now some of the discussion that appears later in the Examples, section, but I felt this deserved to be the in the Theory-part, before explicit examples are discussed.
added now also a section with some pointers to derivators.
The fact that homotopy limit is the first hit presented by Google is something that not so much makes me feel glorious, as that it makes me feel worried about the long distance this entry has from perfection… But I guess I should relax.
I think the page is pretty good. The main problem of organization I see is that there is some duplication in the discussion of projective/injective model structures between the section global definition and the section general formula. Also I’d like to see a mention of Reedy model structures as well; maybe I’ll add that if I get the chance.
Is the section on simplicial presheaves really appropriate on the general homotopy limit page? Would it make more sense at simplicial presheaf or a separate page of its own?
I agree. Yes, the section on simplicial presheaves would be better kept at the entry on simplicial presheaves themselves.
I can move it later this week. Unless you beat me to it.
Unless you beat me to it.
Urs Schreiber starring in a horror movie: "Beaten by nlabizens".
Added statement of proof of the standard fact that every simplicial set is the “hocolim over its cells” to homotopy colimit – examples – hocolims of simplicial diagrams
The query at homotopy limit seems outdated. So I move it here as the archived version:
Tim Mike, are you intending to treat the case of when the domain category, $D$ is the above, is enriched as well? This would handle the example of homotopy limts of homotopy coherent diagrams, both in Vogt’s sense and in the simplicially enriched case looked at by Bourn and Cordier. This would also allow the $G$ in one of the examples to be a simplicial or topological group, or to be (?) and A-infinity category. (Some of those examples may be already dealt with in others of the entries as different people classify things in different ways.)
Perhaps some of the more classical referencs, Vogt, Bousfield-Kan etc. might be included for completeness.
Mike: Yes, certainly; my paper referenced above deals with the case when $D$ is enriched as well. There are cofibrancy technicalities, of course. I’m not against including the classical references, although I find them fairly impenetrable myself.
Thanks, Zoran, for you effots of cleaning up old discussion in the entries and reviving them here. I think this is a very good service.
I added the link to Maltsiniotis’ lectures at Sevilla. Notice that in lecture II, in the setting of a localizer $(C,W)$, he defines a homotopy (co)limit as an adjoint to a localized version $\bar{\Delta}^I: W^{-1} C \to W_I^{-1} C^I$ of the diagonal functor $\Delta^I: C\to C^I$ on the localized category of $I$-diagrams; to start with one does not ask even for 2 out of 3 property for weak equivalences nor for any kind of enrichment. (He neglects the size questions, which are usually resolvable only under more specific assumptions.) See lecture I, localizers pdf pages 12-13.
Added to homotopy limit:
Alternative definitions can be formulated at the level of the homotopy category $W^{-1} C$ one defines a localized version $\bar{\Delta}^I : W^{-1} C\to W_I^{-1} C^I$ of the diagonal functor $\Delta^I : C\to C^I$ and define the homotopy limits and colimits as the adjoints of $\bar{\Delta}^I$ (at least at the points where the adjoints are defined). Here $W_I\subset Mor(C^I)$ are the morphisms of diagrams whose all components are in $W\subset Mor(C)$. The above definitions via derived functors (Kan extensions) follow once one applies the general theorem that the derived functors of a pair of adjoint functors are also adjoint and noticing that $(\Delta^I,\bar{\Delta}^I)$ is a morphism of localizers (and in particular that $\bar{\Delta}^I$ with the identity 2-cell is a Kan extension (simultaneously left and right)).
The previous paragraphs seemed a bit confused about adjoints versus Kan extensions, so I fixed them.
added to homotopy colimit very briefly the statement that the inclusion $\Delta_{only faces} \hookrightarrow \Delta$ is homotopy final
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