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I fixed some formatting.
So what are they for?
I quote the abstract of one of Simon Henry’s papers:
We introduce a notion of “weak model category” which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category. The main advantages of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems for them to form a weak model category. The theory is developed in a very weak constructive framework and we use it to produce, completely constructively (even predicatively), weak versions of various standard model categories, including the Kan-Quillen model structure, the variant of the Joyal model structure on marked simplicial sets, and the Verity model structure for weak complicial sets. We also construct semi-simplicial versions of all these.
I have not looked at the paper at all, and have no opinion on the contents, although the abstract does look quite interesting.
Sure, there is the references cited. But what David C. probably meant, certainly what I would mean, is that a sentence or two on the purpose of the definition is missing in the entry.
Thanks.
The second paragraph of the Idea section, as of now, may need adjustment. It reads:
In particular, one can construct left and right Bousfield localizations in this setting without assuming the model category to be left proper respectively right proper.
Since “this setting” must be that of weak model categories, should “assuming the model category…” rather be “assuming the weak model category…”?
And even with that, I feel something is missing: Continuing from the previous sentence and following “In particular” we need to first state examples of constructions of model category theory that exists also in the weak case. Then as an addeddum to “in particular” we can list extra advantages gained, such as no need for properness.
Thanks, Dmitri, thanks David. But allow me to make a suggestion for how to phrase it:
The concept of weak model categories is a relaxation of that of model categories, even weaker than the concept of semimodel categories, but such that it still allows for a rich theory largely analogous to that of actual model categories:
The weak analogue of the construction of the homotopy category of a model category still exists, as do notions of Quillen adjunction and Quillen equivalence.
Also, for example, an analogue of left or right Bousfield localization of model categories still makes sense for weak model categories; and, as a bonus in contrast to the usual case, it does not require the assumption of left or right properness.
added the publication data for
that just came in. Also fixed the hyperlinking of the references in the text.
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