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tried to polish a little and slightly expand model category, starting with the Definition-section and ending with the (new and tiny) Properties section. Added some more subsections and so on.
I have added to model category a section Redundancy in the axioms with a statement of the fact that a model structure is determined by its cofibrations and (just) the fibrant objects. Included a pointer to a proof by Joyal.
I have now spelled out in the entry the (easy) proof of the above assertion.
I have spelled out the proof that the non-standard definition used in the entry implies that weak equivalences are closed under retracts here. Also added more explicit pointers to Joyal’s text for this statement.
Thanks! This is good to have recorded.
At model category it said
to obtain a quasicategory, given a model category $M$, the simplicial nerve $N_\Delta(M_{cf})$ of the subcategory $M_{cf}\subset M$ of cofibrant-fibrant objects is a quasicategory.
John Baez pointed out to me by email that this is wrong; we need the homotopy coherent nerve of the simplicial category of fibrant-cofibrant objects (which I presume is what whoever wrote this sentence had in mind). I fixed the page.
The definition of model category in the page model category doesn’t include the condition that acyclic cofibrations have the left lifting property with respect to fibrations and cofibrations have the left lifting property with respect to acyclic fibrations. Is this right? Later in the article it says
It is clear that:
Remark
Given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:
given $W$ and $C$, we have $F = RLP(W \cap C)$;
given $W$ and $F$, we have $C = LLP(W \cap F)$;
given $C$ and $F$, we find $W$ as the class of morphisms which factor into a morphism in $C \cap W$ followed by a morphism in $F \cap W$.
The terms RLP and LLP are not explained.
On a separate note, I’m confused about how the third item above lets us find $W$ given $C$ and $F$, because it’s circular. Is this item attempting to say that $W$ is the unique subcategory such that any morphism in that subcategory factors as a morphism in $C \cap W$ followed by a morphism in $F \cap W$? Or maybe that given $C$ and $F$ there’s a unique model structure for which the weak equivalences have this property?
The condition that acyclic cofibrations have the left lifting property for fibrations, etc., is packaged up in the definition of “weak factorization system” here (condition 2 in Definition 2.1).
A non-circular way of interpreting the description of $W$ is to read $C \cap W$ here as being defined as acyclic cofibrations, meaning morphisms with the left lifting property for fibrations, which is determined once you specify $F$.
Okay, I get it. I think this should be explained better.
Feel free to explain it better! (-:
where “RLP” appears first, I added this remark:
(Here $RLP(S)$ denotes the class of morphisms with the right lifting property against $S$ and $LLP(S)$ denotes the class of morphisms with the left lifting property against $S$.)
Until somebody has the energy to rewrite this entry into something more inviting, I have added the following line at the beginning of the Definition-section:
The following is a somewhat terse account. For a more detailed exposition see at Introduction to Homotopy Theory the section Abstract homotopy theory.
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