Talk:Model category

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Well, a category stuck on this could have saved me work; I translated the German page just now as model category. So a merge is now required. Charles Matthews 18:41, 15 October 2005 (UTC)

As far as I can tell, there is a difference between a model category and a closed model category, though they probably belong in the same article. Marc Harper 11:24, 16 October 2005

Now, what would that be? All I see now is the existence of limits. Is the difference that for closed you need all limits and colimits? Or is it more treacherous than that?
By the way, if you put even an approximate category on a mathematical article (e.g. Category:Algebraic topology) it is picked up by a bot and shows up on the page that now links to this article, of recent additions in mathematics. Charles Matthews 09:21, 17 October 2005 (UTC)
The issue is a little ambiguous; the differences are (1) finite limits/colimits versus completeness/cocompleteness and (2) closure of the distinguished classes under composition. The class of weak equivalences is always closed under composition by the 2 of 3 axiom, but not all authors seem to require closure for fibrations/cofibrations. (3) It appears that the classes are not always required to have identity morphisms. See [1] for instance, p4-5. To make the issue worse, closed model categories are often simply referred to as model categories.
Since the difference is not completely clear, I have chosen to stick with the more common and more clearly defined 'closed model category'. Dwyer and Spalinski define a closed model category as this article now reads. I changed all limits/colimits to only the finite limts/colimits. We can merge this article into the model category article, appending additional stipulations for a model category to be closed, redirecting this article there, or vice versa. Lacking a reference that clearly spells out the difference, perhaps it's best to keep it the way it is for now.
This article needs a lot of work, particularly in the definitions of homotopy and all that, which I intend to do, but I will wait to make additional changes until a decision on the merge issue is reached.
I apologize for not properly categorizing the article; I will be sure to include this in the future. Marc Harper 15:55, 19 October 2005

So, I think this should appear on the model category page. There would be no objection to defining first closed model category, and only then weakening axioms - provided that excludes no example that is of major interest. Which brings me round to asking, what sort of range of examples of interest are we talking about here? Charles Matthews 21:11, 19 October 2005 (UTC)

I just checked Hovey's book: he says that the distinction between the two has not proven significant and that most recent authors use closed model categories but drop the adjective closed. So I suggest we do the same, following his lead. He defines a model structure which consists of the axioms but no composition closure properties and then defines a model category as a category with a model structure that also has all small limits and colimits. Apparently there are no major examples where this is an issue, considering what is normally thought of as a space, specifically: topological spaces and simplicial _______ (whatever); as well as other examples such as chain complexes of R-modules or sheaves.

Let's move this to model category, redirect closed model category to model category, and give some explanation in line with Hovey. -- Marc Harper 16:25, 19 October 2005 (CT)

OK. Cut-and-paste this into model category and redirect closed model category there, for the avoidance of doubt. Charles Matthews 21:43, 19 October 2005 (UTC)

Contents

[edit] Move

I actually did it a different way, that puts the CMC page history here. Below is the old model category text, to edit in at need (translation from de:Modellkategorie. Charles Matthews 08:40, 20 October 2005 (UTC)

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows'): 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes (derived category theory). This concept was introduced in 1967 by Daniel G. Quillen.

[edit] Definition

In the category \mathcal{C} are three subcategories defined, of

  • weak equivalences
  • fibrations
  • cofibrations

all having the same collection of objects as C. Fibrations or cofibrations that are weak equivalences are trivial or acyclic

The model category axioms are the following.

[edit] MC1 ((Co-)limits)

\mathcal{C} is finitely complete and cocomplete.

[edit] MC2 ("2 or 3")

If any two out of f, g, gf are weak equivalences in C, so is the third.

[edit] MC3 (Retract)

If f is a retract of g, lying in one of the subcategories, then so does f.

[edit] MC4 (Lifting)

For a commutative diagram

Bild:Hebung.png

with i a cofibration, p a fibration and i or p acyclic, there is an arrow h\colon B\to X, that makes the diagram commute.

[edit] MC5 (Decomposition)

  1. Any arrow can be written p\circ i with a fibration p and an acyclic cofibration i.
  2. Any arrow can be written p\circ i with an acyclic fibration p and a cofibration i.

[edit] Properties

  • The definition is self-dual: the opposite category \mathcal{C}^{op} has a similar structure, if we swap fibration and cofibration over.
  • MC4 picks out the fibrations: an arrow p is a fibration, if and only if for each diagram, in which i is an acyclic cofibration, there is a lifting h (and also for cofibrations, mutandis mutatis). To specify a model category it is therefore sufficient to pick out the weak equivalences and one class out the other two kinds.

[edit] References

  • W. G. Dwyer und J. Spalinski: Homotopy Theories and model categories, 1995 [2]
  • Mark Hovey: Model Categories, 1999, ISBN 0-8218-1359-5
  • Daniel G. Quillen: Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967.de:modellkategorie
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