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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.
Thanks. I added a comment about the Reedy model structure which “explains” the first condition.
Ah, right. Thanks!
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 ?
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.
At homotopy pullback I have added a brief section In homotopy type theory with just a basic remark. Should be expanded, eventually.
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.
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”)?
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.
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.
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)?
Oh thanks, I see now, the argument is in the proof of Lemma 1 in that version. I’ll try to make some edits.
Thanks!
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.
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.
added full publication details for the following item (wasn’t so easy to find…)
just a propos, since it came up in a discussion:
Does anyone have a good counter-example of a pullback of a fibration between fibrant objects to a non-fibrant domain being a homotopy pullback?
Such a counter-example would be good to add to this Prop. to see that its assumptions are necessary.
noticed that “homotopy pushout” has been redirecting to homotopy pullback all along, which now makes it hard to cross-link more specifically for instance to homotopy pushout type.
So hereby I am deleting all pushout related redirects
[[!redirects homotopy pushout]]
[[!redirects homotopy pushouts]]
[[!redirects homotopy co-fiber product]]
[[!redirects homotopy co-fiber products]]
[[!redirects homotopy cobase change]]
[[!redirects homotopy cobase changes]]
and am making them instead point to a new entry homotopy pushout, which i will create now
Noticed that all the (couple of) items in the section “References – In HoTT” (here) were about homotopy pushouts instead of pullbacks.
So I moved this previous content of this subsection to homotopy pushout type and instead added here pointers on homotopy pullbacks in HoTT:
Univalent Foundations Project, Exc. 2.11 in: Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Egbert Rijke, Homotopy pullbacks [pdf], Lecture 10 in: Introduction to Homotopy Type Theory, lecture notes, CMU (2018) [pdf, webpage]
added a list of pointers to textbook chapters dedicated to introducing homotopy pullbacks:
Jeffrey Strom, Homotopy Pullback Squares, §7.3 in: Modern classical homotopy theory, Graduate Studies in Mathematics 127, American Mathematical Society (2011) [doi:10.1090/gsm/127]
Martin Arkowitz, Homotopy Pushouts and Pullbacks, §6 in: Introduction to Homotopy Theory, Springer (2011) [doi:10.1007/978-1-4419-7329-0]
Peter May, Kate Ponto, Some basic homotopy limits: §2.2 in: More concise algebraic topology, University of Chicago Press (2012) [ISBN:9780226511795, pdf]
and took the liberty of pointing also to:
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