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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 30th 2010
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   Author: Harry Gindi
   Format: MarkdownItexIn [this paper](http://www.math.harvard.edu/~clarkbar/complete.pdf) by Clark Barwick, he gives a definition of a $\lambda$-tractable left (resp. right) E-model category as as a left (resp. right) E-model category equipped with two small sets of arrows $I,J$ such that the source and target of all such arrows are $\lambda$-presentable and E-cofibrant (resp. just $\lambda$-presentable) and satisfying the following two conditions: 1.) A morphism (resp. with E-fibrant target) has the rlp with respect to all morphisms in I if and only if it is a trivial fibration. 2.) A morphism has the rlp with respect to all morphisms in J if and only if it is a fibration. This seems a little strange, since up until this point, all of the definitions were obviously dual to one another, but in this case, they seem to just be different. Is there any obvious reason involving duality that explains why this is the case?

   In this paper by Clark Barwick, he gives a definition of a $\lambda$-tractable left (resp. right) E-model category as as a left (resp. right) E-model category equipped with two small sets of arrows $I,J$ such that the source and target of all such arrows are $\lambda$-presentable and E-cofibrant (resp. just $\lambda$-presentable) and satisfying the following two conditions:

   1.) A morphism (resp. with E-fibrant target) has the rlp with respect to all morphisms in I if and only if it is a trivial fibration.

   2.) A morphism has the rlp with respect to all morphisms in J if and only if it is a fibration.

   This seems a little strange, since up until this point, all of the definitions were obviously dual to one another, but in this case, they seem to just be different. Is there any obvious reason involving duality that explains why this is the case?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 30th 2010
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   Author: Urs
   Format: MarkdownItexI am not sure which asymmetry you mean. The notion of [[tractable model category]] is a variant of [[combinatorial model category]], which is in particular a refinement of [[cofibrantly generated model category]]. The (acyclic-)fibration-definition you just mentioned are just those of cofibrantly generated model categories. Is the asymmetry you mean the special role of coifibrations in that definition?

   I am not sure which asymmetry you mean. The notion of tractable model category is a variant of combinatorial model category, which is in particular a refinement of cofibrantly generated model category.

   The (acyclic-)fibration-definition you just mentioned are just those of cofibrantly generated model categories. Is the asymmetry you mean the special role of coifibrations in that definition?

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 30th 2010
   • (edited Jul 30th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexNo no no. Look, these are not honest model categories. A left E-model category is a pair of adjoint functors $F:E\rightleftarrows C:U$ between structured homotopical categories ([[homotopical category|homotopical categories]] with fixed lluf subcategories called fibrations and cofibrations such that cofibrations are stable under pushout and fibrations are stable under pullback) such that $U$ preserves fibrations and trivial fibrations. Further, we require that the initial object of $C$ is $E$-cofibrant, and that any cofibration of $C$ with $E$-fibrant source is an $E$-cofibration (We call a morphism $f$ in $C$ and $E$-cofibration if $U(f)$ is a cofibration in $E$ (and an object $a$ is E-cofibrant if U(a) is E-cofibrant).). Even further, we require that cofibrations have the llp with respect to trivial fibrations, and that fibrations have the rlp with respect to cofibrations with $E$-cofibrant domain. Lastly, we require that any morphism has a functorial factorization as a cofibration and a trivial fibration, and that any morphism with $E$-cofibrant source has a factorization as a trivial cofibration and a fibration. A right $E$-model category is a pair of adjoint functors $U:C\rightleftarrows E:F$ such that the opposite adjunction $F^{op}:E^{op}\rightleftarrows C^{op}:U^{op}$ is a left $E^{op}$-model category. Now the defintion up top should make some more sense. I didn't write $E$-cofibration for my health! With regard to my original question, I think that a careful analysis should yield that the left $E^{op}$-model category assigned to a right tractable $E$-model category is necessarily tractable, because otherwise, it's not clear to me why they should even be called the same thing.

   No no no. Look, these are not honest model categories. A left E-model category is a pair of adjoint functors $F:E\rightleftarrows C:U$ between structured homotopical categories (homotopical categories with fixed lluf subcategories called fibrations and cofibrations such that cofibrations are stable under pushout and fibrations are stable under pullback) such that $U$ preserves fibrations and trivial fibrations. Further, we require that the initial object of $C$ is $E$-cofibrant, and that any cofibration of $C$ with $E$-fibrant source is an $E$-cofibration (We call a morphism $f$ in $C$ and $E$-cofibration if $U(f)$ is a cofibration in $E$ (and an object $a$ is E-cofibrant if U(a) is E-cofibrant).). Even further, we require that cofibrations have the llp with respect to trivial fibrations, and that fibrations have the rlp with respect to cofibrations with $E$-cofibrant domain. Lastly, we require that any morphism has a functorial factorization as a cofibration and a trivial fibration, and that any morphism with $E$-cofibrant source has a factorization as a trivial cofibration and a fibration.

   A right $E$-model category is a pair of adjoint functors $U:C\rightleftarrows E:F$ such that the opposite adjunction $F^{op}:E^{op}\rightleftarrows C^{op}:U^{op}$ is a left $E^{op}$-model category.

   Now the defintion up top should make some more sense.

   I didn’t write $E$-cofibration for my health!

   With regard to my original question, I think that a careful analysis should yield that the left $E^{op}$-model category assigned to a right tractable $E$-model category is necessarily tractable, because otherwise, it’s not clear to me why they should even be called the same thing.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 30th 2010
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   Author: Urs
   Format: MarkdownItex> No no no. Look, Are you trying to motivate me to make a second attempt to interpret your question? What is your question, Harry?

   No no no. Look,

   Are you trying to motivate me to make a second attempt to interpret your question?

   What is your question, Harry?

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 30th 2010
   • (edited Jul 30th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThis notion is significantly more complicated in the case of "left and right model categories", which are not actually model categories. I'm sorry if I came off a bit rude. It's just that the nLab's definition does not cover this generalization. The asymmetry I'm talking about is pretty clear just from looking at the definition (completely different things need to be E-fibrant and E-cofibrant), once you're looking at left and right E-model categories.

   This notion is significantly more complicated in the case of “left and right model categories”, which are not actually model categories. I’m sorry if I came off a bit rude. It’s just that the nLab’s definition does not cover this generalization.

   The asymmetry I’m talking about is pretty clear just from looking at the definition (completely different things need to be E-fibrant and E-cofibrant), once you’re looking at left and right E-model categories.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 30th 2010
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   Author: Mike Shulman
   Format: MarkdownItexI would guess that it probably has to do with the fact that lots of model categories arising "in nature" are cofibrantly generated, while few are fibrantly generated.

   I would guess that it probably has to do with the fact that lots of model categories arising “in nature” are cofibrantly generated, while few are fibrantly generated.