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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexprompted by a question by email, I have expanded at [[homotopy pullback]] the section on [Concrete constructions](http://ncatlab.org/nlab/show/homotopy+pullback#ConcreteConstructions) by listing and discussing the precise conditions under which ordinary pullbacks are homotopy pullbacks. Most of this information is scattered around elswehere on the $n$Lab (such as at [[homotopy limit]] and [[right proper model category]]) and I had wrongly believed that it was already collected here. But it wasn't.

   prompted by a question by email, I have expanded at homotopy pullback the section on Concrete constructions by listing and discussing the precise conditions under which ordinary pullbacks are homotopy pullbacks.

   Most of this information is scattered around elswehere on the $n$Lab (such as at homotopy limit and right proper model category) and I had wrongly believed that it was already collected here. But it wasn’t.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks. I added a comment about the Reedy model structure which "explains" the first condition.

   Thanks. I added a comment about the Reedy model structure which “explains” the first condition.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 9th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, right. Thanks!

   Ah, right. Thanks!

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeApr 15th 2012
   • (edited Apr 15th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThere is a sentence in the middle > The canonical morphism $A \times_{C} B \to A \times_C^h B$ here is induced by the [[section]] $Z \to Z^I$. Not only that there is some apparent notation confusion here, but I do not know which canonical morphism, I mean the ordinary pullback $A\times_C B$ is not mentioned in the preceding diagrams. Well, it is, in a different notation, the whole page scrolled above, in a different section. What is going on ?

   There is a sentence in the middle

   The canonical morphism $A \times_{C} B \to A \times_C^h B$ here is induced by the section $Z \to Z^I$.

   Not only that there is some apparent notation confusion here, but I do not know which canonical morphism, I mean the ordinary pullback $A\times_C B$ is not mentioned in the preceding diagrams. Well, it is, in a different notation, the whole page scrolled above, in a different section. What is going on ?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 15th 2012
   • (edited Apr 15th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe _zee_ is a _cee_ :-) The morphism is induced from the canonical section $s : C \to C^I$ into the path space object, namely from the commuting diagram $$ \array{ A \times_C B &\to& C &\stackrel{s}{\to}& C^I \\ \downarrow &&&{}_{\mathllap{\Delta}}\searrow& \downarrow \\ A \times B &&\to&& C \times C } \,. $$ Thanks for catching this. I have fixed it.

   The zee is a cee :-)

   The morphism is induced from the canonical section $s : C \to C^I$ into the path space object, namely from the commuting diagram

   $$\array{ A \times_C B &\to& C &\stackrel{s}{\to}& C^I \\ \downarrow &&&{}_{\mathllap{\Delta}}\searrow& \downarrow \\ A \times B &&\to&& C \times C } \,.$$

   Thanks for catching this. I have fixed it.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt _[[homotopy pullback]]_ I have added a brief section _[In homotopy type theory](http://ncatlab.org/nlab/show/homotopy+pullback#InHomotopyTypeTheory)_ with just a basic remark. Should be expanded, eventually.

   At homotopy pullback I have added a brief section In homotopy type theory with just a basic remark. Should be expanded, eventually.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2012
   • (edited May 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexat _[[homotopy pullback]]_ I have added another brief section _[Constructions - In homotopy type theory](http://ncatlab.org/nlab/show/homotopy+pullback#ConstructionInHomotopyTypeTheory)_ where I spell out briefly how applying the categorical semantics interpretation to the HoTT formulation of the homotopy pullback does reproduce the derived pullback formula discussed further above in this entry.

   at homotopy pullback I have added another brief section Constructions - In homotopy type theory where I spell out briefly how applying the categorical semantics interpretation to the HoTT formulation of the homotopy pullback does reproduce the derived pullback formula discussed further above in this entry.

8. • CommentRowNumber8.
   • CommentAuthoradeelkh
   • CommentTimeApr 13th 2014
   • (edited Apr 13th 2014)
   • PermaLink
   Author: adeelkh
   Format: MarkdownItexI am having some trouble following the discussion between Remark 1 and Corollary 1 at [[homotopy pullback]], > If all objects involved are already fibrant, then such a resolution is provided by the [[factorization lemma]]. This says that a fibrant resoltuion of $B \to C$ is given by the total composite vertical morphism in > > $$ \array{ C^I \times_C B &\to& B \\ \downarrow && \downarrow \\ C^I &\to& C \\ \downarrow \\ C } \,, $$ > > where $C \stackrel{\simeq}{\to} C^I \to C \times C$ is a [[path object]] for the [[fibrant object]] $C$ With a friend we were able to see that this vertical composite $C^I \times_C B \to C$ is a fibration by factoring it as two fibrations, $$C^I \times_C B \to (C \times C) \times_C B \simeq C \times B \to C \times \ast$$ After some reflection one can see that this is the same as the morphism in question. However I wonder if the author maybe had a more elegant argument in mind (I don't really understand the reference to the "factorization lemma")?

   I am having some trouble following the discussion between Remark 1 and Corollary 1 at homotopy pullback,

   If all objects involved are already fibrant, then such a resolution is provided by the factorization lemma. This says that a fibrant resoltuion of $B \to C$ is given by the total composite vertical morphism in

   $$\array{ C^I \times_C B &\to& B \\ \downarrow && \downarrow \\ C^I &\to& C \\ \downarrow \\ C } \,,$$

   where $C \stackrel{\simeq}{\to} C^I \to C \times C$ is a path object for the fibrant object $C$

   With a friend we were able to see that this vertical composite $C^I \times_C B \to C$ is a fibration by factoring it as two fibrations,

   $$C^I \times_C B \to (C \times C) \times_C B \simeq C \times B \to C \times \ast$$

   After some reflection one can see that this is the same as the morphism in question. However I wonder if the author maybe had a more elegant argument in mind (I don’t really understand the reference to the “factorization lemma”)?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, this is supposed to be the key argument laid out at _[[factorization lemma]]_. Just recently Richard Williamson had rewritten that entry. See our recent discussion around [here](http://nforum.mathforge.org/discussion/1653/factorization-lemma/?Focus=45946#Comment_45946). If the above doesn't become clear enough there, then we should add more amplification again.

   Hm, this is supposed to be the key argument laid out at factorization lemma.

   Just recently Richard Williamson had rewritten that entry. See our recent discussion around here.

   If the above doesn’t become clear enough there, then we should add more amplification again.

10. • CommentRowNumber10.
    • CommentAuthoradeelkh
    • CommentTimeApr 13th 2014
    • PermaLink
    Author: adeelkh
    Format: MarkdownItexI can't really make the connection, is it Proposition 3 which is supposed to be used? I thought that was just the analogue of the factorization axiom for categories of fibrant objects.

    I can’t really make the connection, is it Proposition 3 which is supposed to be used? I thought that was just the analogue of the factorization axiom for categories of fibrant objects.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, so maybe we need to add more to Richard's version then after all. If you go back to [revision number 10](http://ncatlab.org/nlab/revision/factorization+lemma/10) (or any previous one) then the statement in question is right in the first paragraphs. If that's what you need, maybe could you edit the entry again such as to make this clear (again)?

    Okay, so maybe we need to add more to Richard’s version then after all.

    If you go back to revision number 10 (or any previous one) then the statement in question is right in the first paragraphs.

    If that’s what you need, maybe could you edit the entry again such as to make this clear (again)?

12. • CommentRowNumber12.
    • CommentAuthoradeelkh
    • CommentTimeApr 13th 2014
    • PermaLink
    Author: adeelkh
    Format: MarkdownItexOh thanks, I see now, the argument is in the proof of Lemma 1 in that version. I'll try to make some edits.

    Oh thanks, I see now, the argument is in the proof of Lemma 1 in that version. I’ll try to make some edits.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks!

    Thanks!

14. • CommentRowNumber14.
    • CommentAuthoradeelkh
    • CommentTimeApr 14th 2014
    • (edited Apr 14th 2014)
    • PermaLink
    Author: adeelkh
    Format: MarkdownItexOk, I've made some changes to [[factorization lemma]] now, mainly adding lemma 3 and rewriting the proof of the factorization lemma (prop 1). I hope everything is still coherent. Edit: also edited [[homotopy pullback]] a bit. I changed the limit in Corollary 1 to the pullback you see now, just for simplicity of exposition.

    Ok, I’ve made some changes to factorization lemma now, mainly adding lemma 3 and rewriting the proof of the factorization lemma (prop 1). I hope everything is still coherent.

    Edit: also edited homotopy pullback a bit. I changed the limit in Corollary 1 to the pullback you see now, just for simplicity of exposition.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2014
    • (edited Apr 30th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexhave added to the section _[Fiberwise characterisation](http://ncatlab.org/nlab/show/homotopy+pullback#HomotopyFiberCharacterization)_ the statement for stable model categories: a square is a homotopy pullback iff it is a homotopy pushout iff it induces an equivalence on homotopy fibers. With a pointer to the 2007 version of Hovey's book, for a proof.

    have added to the section Fiberwise characterisation the statement for stable model categories: a square is a homotopy pullback iff it is a homotopy pushout iff it induces an equivalence on homotopy fibers. With a pointer to the 2007 version of Hovey’s book, for a proof.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded full publication details for the following item (wasn't so easy to find...) * [[Wojciech Chacholski]], Wolfgang Pitsch, and [[Jerome Scherer]], _Homotopy pullback squares up to localization_, in [[Dominique Arlettaz]], [[Kathryn Hess]] (eds.) _An Alpine Anthology of Homotopy Theory_ ([arXiv:math/0501250](http://arxiv.org/abs/math/0501250), [pdf](http://sma.epfl.ch/~jscherer/articles/hopullbacks2.pdf), [doi:10.1090/conm/399](http://dx.doi.org/10.1090/conm/399)) <a href="https://ncatlab.org/nlab/revision/diff/homotopy+pullback/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/homotopy+pullback/50">v50</a>, <a href="https://ncatlab.org/nlab/show/homotopy+pullback">current</a>

    added full publication details for the following item (wasn’t so easy to find…)

    • Wojciech Chacholski, Wolfgang Pitsch, and Jerome Scherer, Homotopy pullback squares up to localization, in Dominique Arlettaz, Kathryn Hess (eds.) An Alpine Anthology of Homotopy Theory (arXiv:math/0501250, pdf, doi:10.1090/conm/399)

    diff, v50, current