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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 23rd 2009
• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeNov 23rd 2009

I redirected it from model structure on an under category.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 23rd 2009

By the way: I keep seeing in the literature overcategory instead of over category . For instance in the article by Hirschhorn linked to at model structure on an over category.

Are we sure we want to have the entries named over category and so on?

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeNov 23rd 2009

Well, I like slice category, but I remember putting it over category in the days before redirects to help insure that your links to it would work.

I have put in redirects for overcategory and the like.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 23rd 2009

Oh, really, was it me who started writing "over category"? I forget. Sorry, then! :-)

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeDec 4th 2009

Now I am interested in the special case of Top with Strom's model structure. Then there are theorems on the connection between Dold fibrations and Hurewicz fibrations, one of them is that every Dold fibration p:E -> B is

   homotopy equivalent over B


with a Hurewicz fibration p:E' to B. Is this giving some light on the open question as if there is a model category structure on Top where fibrations are Dold fibrations ? Another important thing is that you can verbatim repeat the definition of Hurewitz fibration to get Dold fibration if instead of homotopies you use delayed homotopies (this is a theorem). Is there a way to use delayed homotopies to nontrivially modify the notion of cofibration ?

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeDec 4th 2009
In the case of the under category, a relevant theorem may be Dold's theorem which states that a map whose underlying map is a homotopy equivalence is already a homotopy equivalence under provided its source and target are cofibrations. This is discussed in Kamps-Porter in quite a lot of detail.
• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeDec 7th 2009

These are still usual cofibrations, and these do not form a model category with Dold fibrations, but maybe there is a good modified choice of cofibrations which woudl be "complementary" with Dold fibrations (maybe silly idea knowing something specific banning this choice, but to me it looks still reasonable).

• CommentRowNumber9.
• CommentAuthordomenico_fiorenza
• CommentTimeJan 3rd 2010
• (edited Jan 3rd 2010)
modified Idea in over quasi-category. now it should be less evil.
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJan 4th 2010

That reminds me: we should add a discussion about if and how the model structure on an over category models the corresponding over quasi-category. I was about to make the obvious statement, but I'll need to check something first.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMar 18th 2011

I have somewhat hastily added to model structure on an over category the argument that over a fibrant object this presents the correct over-$(\infty,1)$-category.

However, I have to dash off now and go offline. Will try to look into this again later.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMar 18th 2011

have now found a few minutes to expand and polish the proof

• CommentRowNumber13.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 21st 2016

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeApr 18th 2017

Hm, the entry slice model structure states that slicing preserves cofibrant generation, properness, combinatoriality, but then what it means to use is preservation of simplicial model structure…

• CommentRowNumber15.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 7th 2017

Re #14: I added Proposition 2.3, which shows that if C is a simplicial model category, then so is C/X.