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A stub for Cartan-Eilenberg categories.
I just wrote a Zbl review on one of these papers so have added a short definition (still stubby) of Cartan-Eilenberg categories.
I have added some hyperlinks.
Could you maybe clarify and/or expand on the sentence
This allows a neat development of a new form of homotopical algebra.
?
Isn’t it rather just a new tool for homotopical algebra?
(By the way, it is maybe unwise to say in an entry that something is “recent”, unless you are sure you’ll stay around looking after updating that entry as time goes by…)
The first glitch is mine and was a copy and paste which should not have been included. OOPS!
I have two of the papers so may add in more later. I do not know what the real definition of homotopical algebra is so I am not 100% sure how to call this idea. The recent (and there is a time stamp on this comment so I can use that here!:-)) paper by Pascual not only links with Sullivan minimal models but also with finite topological spaces as studied buy Barmak and Minian.
I suggest we remove the hyperlink to cofibrant as the notion is possibly different here than in the usual sense. (Edit:Rather than change the link I added a few words to that effect.)
Tim,
I suppose you want in particular to obtain the ordinary localization at the strong and weak equivalences, right? If so, that’s the ordinary homotopy theory of the category with weak equivalences, produced with another tool in homological algebra, I would say.
But I don’t want to be hair-splitting, I just would like to nudge you to give other readers, not me, a clear(er) idea of what “new form of homotopical algebra” might possibly refer to, for it sounds a bit mysterious.
The ’new form’ has been deleted!!!! I need to read the older of the two papers again and summarise the points. (I had let a backlog of Zbl reviews build up so have been trying to clear them. This means speed reading the papers rather than taking my time…. silly.) The papers do not seem to give a definite answer to the relationship between this approach and others as I remember it.
Okay, thanks. And no rush, I was just wondering.
I was at a workshop in Sevilla few years ago, and there were many good model category theorists there and people agree that Cartan-Eilenberg categories capture some aspect yet not covered in model categorical (and similar) formalisms. But I do not remember the details of the argument.
Sure, model categories are also just one tool for studying the localization of a category at weak equivalences (simplicial localization, generally). This is what homotopical algebra is about, tools for producing (simplicial) localizations.
And this I gather is also what Cartan-Eilenberg categories are a tool for.
Okay, I looked at the article
It says everything right there in the introduction.
So I have now added the following paragraphs to Cartan-Eilenberg category:
The notion of Cartan-Eilenberg categories is a tool in homotopical algebra. The structure of a Cartan-Eilenberg category (SAPR 10) on a category is the structure of a category with weak equivalences equipped with the further data of “strong” equivalences inducing a notion of cofibrant objects, such that the localization of the category at the weak equivalences is equivalently that of the full subcategory of cofibrant objects at just the strong equivalences.
The motivation for this axiomatics in (SAPR 10) is that it is the general context in which the original construction in (Cartan-Eilenberg 56) of derived functors of additive functors between categories of modules generalizes.
Every model category is in particular a Cartan-Eilenberg category, the strong equivalences in this case are the left homotopy equivalences. But the notion of CE-categories is weaker and exists in situations where a full model category structure is not available. In this way the notion of Cartan-Eilenberg category structures is analogous to other such structures in between categories with weak equivalences and fully-fledged model categories, such as categories of fibrant objects, Waldhausen categories etc.
Great. Warning: the “descent category” in the sense of one of the cited papers – it is not the same what Street calls descent category (and the older tradition calls descriptively more precisely the category of descent data), though it has some relation to descent.
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