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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 8th 2010
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   Author: Mike Shulman
   Format: MarkdownEmily Riehl created [[natural weak factorization system]].

   Emily Riehl created natural weak factorization system.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 16th 2010
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   Author: Urs
   Format: MarkdownItexThe entry uses the symbols $[2]$ and $[3]$ where I would have expected the symbols $[1]$ and $[2]$. Wouldn't that be more standard?

   The entry uses the symbols $[2]$ and $[3]$ where I would have expected the symbols $[1]$ and $[2]$. Wouldn’t that be more standard?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 16th 2010
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   Author: Urs
   Format: MarkdownItexI would be fine with "$2$" and "$3$" instead of "$[2]$" and "$[3]$".

   I would be fine with “$2$” and “$3$” instead of “$[2]$” and “$[3]$”.

4. • CommentRowNumber4.
   • CommentAuthorEmily Riehl
   • CommentTimeJul 1st 2010
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   Author: Emily Riehl
   Format: TextI wrote this ages ago (perhaps my first n-lab post) and didn't really know what I was doing with the mathematical typesetting. I think 2 or 3 is best. While we're on this topic, Richard Garner, Peter Lumsdaine, and I have been conspiring to change the name from "natural" weak factorization system to "algebraic" weak factorization system, and it seems we've reached an agreement to start using this term in the future. When is the appropriate time to make the chance on the n-lab?

   I wrote this ages ago (perhaps my first n-lab post) and didn't really know what I was doing with the mathematical typesetting. I think 2 or 3 is best.
   
   While we're on this topic, Richard Garner, Peter Lumsdaine, and I have been conspiring to change the name from "natural" weak factorization system to "algebraic" weak factorization system, and it seems we've reached an agreement to start using this term in the future. When is the appropriate time to make the chance on the n-lab?
5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 1st 2010
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   Author: Mike Shulman
   Format: MarkdownItexI would say the appropriate time to change it is after you suggest it on the n-Forum and no one objects for a while. (-: I think it's a fine idea. But I also feel like these things could even use a much snappier name, akin to "monad," especially in view of what Richard was saying about how any accessible monad on a locally presentable category can be made the fibrant replacement monad in a nwfs (oops, awfs?). I think if I had the world to do over again, I might seriously consider calling these things simply "factorization systems" and the older version "unique factorization systems." That would probably create too much confusion if we started doing it now. But nwfs and awfs are both kind of long and cumbersome to say.

   I would say the appropriate time to change it is after you suggest it on the n-Forum and no one objects for a while. (-:

   I think it’s a fine idea. But I also feel like these things could even use a much snappier name, akin to “monad,” especially in view of what Richard was saying about how any accessible monad on a locally presentable category can be made the fibrant replacement monad in a nwfs (oops, awfs?). I think if I had the world to do over again, I might seriously consider calling these things simply “factorization systems” and the older version “unique factorization systems.” That would probably create too much confusion if we started doing it now. But nwfs and awfs are both kind of long and cumbersome to say.

6. • CommentRowNumber6.
   • CommentAuthorspitters
   • CommentTimeOct 17th 2015
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   Author: spitters
   Format: MarkdownItexIs there an established notion of equivalence of [[algebraic weak factorization system]] or of [[category of fibrant objects]], I expect this to be specialization of the notion of [[Quillen equivalence]].

   Is there an established notion of equivalence of algebraic weak factorization system or of category of fibrant objects, I expect this to be specialization of the notion of Quillen equivalence.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2015
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   Author: Mike Shulman
   Format: MarkdownItexI don't know of one. My guess would be that it wouldn't be as useful, since in the absence of colimits, left adjoints are less likely to exist.

   I don’t know of one. My guess would be that it wouldn’t be as useful, since in the absence of colimits, left adjoints are less likely to exist.

8. • CommentRowNumber8.
   • CommentAuthorKarol Szumiło
   • CommentTimeOct 18th 2015
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   Author: Karol Szumiło
   Format: MarkdownItexOn the other hand, there is an established definition of an equivalence of categories of fibrant objects. It is an exact functor (preserving fibrations, acyclic fibrations, terminal object and pullbacks along fibrations) that induces an equivalence of the homotopy categories.

   On the other hand, there is an established definition of an equivalence of categories of fibrant objects. It is an exact functor (preserving fibrations, acyclic fibrations, terminal object and pullbacks along fibrations) that induces an equivalence of the homotopy categories.