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created algebraic model category
The theorem that model category structure plus extra bits on a category induces similar structure on diagram categories was already in Edwards and Hastings lecture notes way back. Does anyone have an idea of the evolution of this result as it seems to be reproved often without reference to them?
Tim, can you give more details on what Edwards and Hastings actually said?
I mean to by adding into the prohomotopy theory in the Lab, but basically they took a category C with a closed model structure, with a condition N: (I will give this in their terms (this is in 1976 so there will be neater ways of stating it, but will copy from my copy which is in front of me.))
N1: Each cofibration is a pushout of a cofibration of cofibrant objects.
N2 Each fibration is a pullback of a fibration of fibrant objects.
N3 either all objects are cofibrant or all objects are fibrant
N4 there exist functorial cylinder objects. (then some notation we don’t need here).
(This is on page 45.)
They then take $C^J$, the category of $J$-indexed diagrams with level cofibrations and weak equivalences for the obvious classes and the fibrations to be those morphisms with RLP wrt trivial cofibs. The additional restriction is that $J$ is a cofinite strongly directed set (p.57.)
This is in Springer LN 542.
I remember in Pursuing Stacks, there was discussion of this sort of result, that Heller did a lot of work on it in his MAS memoir, and so on, then it seems to disappear for a time before reappearing with no mention of Edwards and Hastings, nor of Heller, until that omission is acknowledged. I am just curious what other stages along the way are there. E &H wanted this to get their model cat. structure on ProC so as to mirror proper homotopy theory and to get a strong shape theory, so the restriction on $J$ is natural for them. Did Kan’s school give another variant of this at about the same time? (They produced such a lot at that time that I find it hard to give credit where credit is due.)
The recent work in this area is by Dan Isaksen who calls the E&H structure on Pro-sSet the strict model structure, and then does a neat localisation to get a structure on that category that does what Artin and Mazur seemed to have hoped for in SLN 100, and which is related to the Morel-Voevodsky homotopy theory in a nice way.
Some of this is being put in the monograph on profinite stuff that I have been finishing off (slowly) for some time, so when it is a bit more advanced I will try to put a summary on the nLab. (I started it in 1984 so progress is slow!!!!!)
In the case somebody does not know Edwards/Hastings Springer lecture notes volume is online: pdf.
Thanks, Zoran. I have a hardcopy but it is good to know that there is a lightweight version!!! for when I am away from home. :-)
Prompted by Mike’s recent edits, I’m just wondering whether we should also mention a different sense of ’algebraic’ model structure, of which the folk model structure on groupoids would be a prototypical example. I’m not sure exactly what would one take to be the definition of this sense, but it would be something like: the underlying category is 2-algebraic; everything is both fibrant and cofibrant (although many would I think relax cofibrancy); and the equivalences are those coming from the 2-algebraicity. One point of this is to distinguish between different flavours of the homotopy hypothesis. For example, the Thomason model structure on categories, or the model structures on n-fold categories, are not algebraic in this sense, and this captures one sense in which they are not model structures which satisfy the homotopy hypothesis in a strict sense: although the underlying categories are algebraic, the model structure is not.
Do you mean something like an algebraic definition of higher categories? Maybe we could just expand on that page on the sort of model structures one tends to find; there’s already a remark that usually all objects are fibrant.
Thanks, yes, that sounds like a good suggestion.
Okay.
I don’t think I’m going to do it myself, though. (-:
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