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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 27th 2010

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…)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 26th 2010

added to transferred model structure the (trivial) remark that transfer preserves right properness.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeDec 30th 2018

added citation to Lack, and move “cofibrantly generated” from the definition of transferred model structure to the general theorem about its existence.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeDec 30th 2018

Separated out the proofs of factorization and acyclicity, added a version of the proof of factorization that uses algebraic technology rather than cofibrant generation, and mentioned another non-example (double categories).

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 16th 2019

Added the proof of acyclicity in the presence of path objects.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJan 16th 2019

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)!

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeJan 16th 2019

The other examples of this bug I encountered yesterday are here. Are other people seeing it too?

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeJan 16th 2019

Yes, I’m seeing these problems too.

1. 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 :-).

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeJan 17th 2019

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.

3. Re-rendered to fix Tikz diagrams. Look fine to me now, but please check.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeFeb 13th 2019

Added reference to Makkai-Rosicky, who show that right-lifted wfs preserve combinatoriality.

4. corrected an evident typo

steveawodey

• CommentRowNumber15.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 25th 2022

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJul 20th 2022

I have touched the formatting (and the punctuation) of the definition and the first existence theorems, just in order to make it all a little easier on the eye (hopefully)

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJul 21st 2022

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJul 21st 2022

and its erratum:

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJul 21st 2022
• (edited Jul 21st 2022)

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:

1. $\mathcal{C}$ and $\mathcal{D}$ are locally presentable categories,

2. $\mathcal{C}$ is equipped with the structure of a cofibrantly generated model category (hence a combinatorial model category) with classes of (co-)/fibrations and weak equivalences $Cof, Fib, W \,\subset\, Mor(\mathcal{C})$,

3. $RLP\big( L^{-1} Cof \big) \,\subset\, L^{-1}(W)$ (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?

• CommentRowNumber20.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 21st 2022

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.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJul 21st 2022

Thanks for the sanity check!

I’ll wait a little, but it seems like we should adjust the wording.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeAug 8th 2022

I have tried to polish up the list of references on the existence of right transfer, now starting here.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeAug 8th 2022

and added actual mentioning of the term “Kan-transfer theorem” (why Kan alone?)

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeAug 8th 2022

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTime2 days ago

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.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTime2 days ago
• (edited 2 days ago)

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTime2 days ago
• (edited 2 days ago)

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.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTime2 days ago

So I have now made the argument for the proof of the combinatorial left transfer more explicit – here.

Now that I have dug into this, I realize that this must be what the previous lemma (here) was alluding to (re #20 above).