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added to transferred model structure a simple remark in a subsection Enrichement on conditions that allow to transfer also an enriched model structure.
(The example I am thinking of is transferring the sSet-enriched model structure on cosimplicial rings to one on cosimplicial smooth algebras. But I won’t type that into the entry for the moment…)
added to transferred model structure the (trivial) remark that transfer preserves right properness.
The second diagram in this proof is another garbled one. But when I copy and paste it at the Sandbox it looks fine. This is really weird (and a serious bug)!
The other examples of this bug I encountered yesterday are here. Are other people seeing it too?
Yes, I’m seeing these problems too.
Thanks for raising. Cannot look into it now, but will do at the earliest opportunity. Thanks for trying out the new features, this is the best way to get them into ship shape :-).
Not forgotten about this, but could not get to it today unfortunately.
In light of the now-known fact that any accessible wfs can be both left- and right-transferred, hence any accessible model category can be both left- and right-transferred in the presence of the acyclicity condition of the appropriate handedness, I rewrote this page to be more ambidextrous, specifying “right-transferred” and “left-transferred” as appropriate.
Re-rendered to fix Tikz diagrams. Look fine to me now, but please check.
have added publication data for:
also added publication data for:
and its erratum:
I have added (here) statement of the following theorem (BHKKRS 2015, Thm. 2.23):
Given a pair of adjoint functors $\mathcal{D} \underoverset {\underset{R}{\longleftarrow}} {\overset{L}{\longrightarrow}} {\;\;\bot\;\;} \mathcal{C}$ such that:
$\mathcal{C}$ and $\mathcal{D}$ are locally presentable categories,
$\mathcal{C}$ is equipped with the structure of a cofibrantly generated model category (hence a combinatorial model category) with classes of (co-)/fibrations and weak equivalences $Cof, Fib, W \,\subset\, Mor(\mathcal{C})$,
$RLP\big( L^{-1} Cof \big) \,\subset\, L^{-1}(W)$ (i.e. co-anodyne maps are weak equivalences),
then the left-transferred model category structure on $\mathcal{D}$ exists (i.e. with cofibrations $L^{-1}(Cof)$ and weak equivalences $L^{-1}(W)$) and is itself cofibrantly generated.
By the way, in the References-section it says that this article is about accessible instead of locally presentable transfer, and that the locally presentable left transfer is discussed in HTT – neither of which seems to be the case – am I missing something?
By the way, in the References-section it says that this article is about accessible instead of locally presentable transfer, and that the locally presentable left transfer is discussed in HTT – neither of which seems to be the case – am I missing something?
Added in Revision 28 by Mike Shulman on February 13, 2019.
As far as I can see, the only left-transferred model structure discussed by Lurie is the injective model structure, which is constructed using the Smith recognition theorem.
The same revision also claims that Makkai–Rosický discuss left transfer in their paper Cellular categories, but I am unable to find anything about left transfer there.
Thanks for the sanity check!
I’ll wait a little, but it seems like we should adjust the wording.
added pointer also to:
and added actual mentioning of the term “Kan-transfer theorem” (why Kan alone?)
added pointer to:
I was about to add statement about the left tranferred model structure here being itself cofibrantly generated. But looking at Thm. 2.23 (p. 9) of BHKKRS15 it seems confusing: The statement announces cofibrant generation, but then the proof checks fibrant generation. No?
Maybe I am misreading something. Have to interrupt now, will try again later.
added pointer to:
Re #25:
Oh, I see: the wording in that Thm.+Proof 2.23 is indeed a little off, but the cofibrant generation which the theorem advertises is part of the result by Makkai & Rosický that is being referenced (their Thm. 3.2 on p. 7 together with the later Rem. 3.8):
A generating set of the left transferred cofibrations is given by those between $\kappa$-presentable objects.
Hm, that’s weaker than I was hoping for: One would hope that the pre-images of the generating cofibrations in the codomain are at least “close to” being generating cofibrations of the transferred structure.
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