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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 28th 2010

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 7th 2012
• (edited Mar 21st 2012)

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 21st 2012

I have now spelled out in the entry the (easy) proof of the above assertion.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 8th 2016

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.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJul 8th 2016

Thanks! This is good to have recorded.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeAug 2nd 2017

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.

• CommentRowNumber7.
• CommentAuthorJohn Baez
• CommentTimeDec 4th 2018
• (edited Dec 4th 2018)

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?

• CommentRowNumber8.
• CommentAuthorRuneHaugseng
• CommentTimeDec 4th 2018

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 $F$.

• CommentRowNumber9.
• CommentAuthorJohn Baez
• CommentTimeDec 4th 2018

Okay, I get it. I think this should be explained better.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeDec 5th 2018

Feel free to explain it better! (-:

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeDec 5th 2018

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 $S$.)

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeDec 5th 2018
• (edited Dec 5th 2018)

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.

• CommentRowNumber13.
• CommentAuthorMarc
• CommentTimeDec 6th 2018

Just changed the explanation as suggested in #8