Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 25th 2010
   • (edited Mar 25th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownSo I was reading through Goerss-Jardine, and I realized that Lurie defines a model category in the HTT appendix to be a closed model category. To check this, I looked it up in Quillen's _Homotopical Algebra_, and indeed, Lurie requires that all model categories are closed. The nLab does this as well at [[model category]]. If we use the standard definition of a not-necessarily-closed model category, h(C) is not the Bousfield localization, it is the quotient by the relation of left-right homotopy (using cylinder and path objects). If we assume all model categories are closed, the standard model structure on topological spaces is not a model structure at all, since weak equivalences are not always homotopy equivalences. Has there been a change in terminology, or is there a reason why Lurie made this decision? Edit: I found a discussion in Hovey. We should probably add something to the nLab page.

   So I was reading through Goerss-Jardine, and I realized that Lurie defines a model category in the HTT appendix to be a closed model category. To check this, I looked it up in Quillen's Homotopical Algebra, and indeed, Lurie requires that all model categories are closed. The nLab does this as well at model category.

   If we use the standard definition of a not-necessarily-closed model category, h(C) is not the Bousfield localization, it is the quotient by the relation of left-right homotopy (using cylinder and path objects). If we assume all model categories are closed, the standard model structure on topological spaces is not a model structure at all, since weak equivalences are not always homotopy equivalences.

   Has there been a change in terminology, or is there a reason why Lurie made this decision?

   Edit: I found a discussion in Hovey. We should probably add something to the nLab page.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 25th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownI don't recall what exactly Hovey writes. But I think it is common current practice to not say "Quillen closed model category" but just model category. First of all this is the case that matters. Second the "closed" very badly collides with the notion of [[closed category]]. But, yes, we should at least add a remark on terminology. I don't have the time. Would very much appreciate if you add something.

   I don't recall what exactly Hovey writes. But I think it is common current practice to not say "Quillen closed model category" but just model category.

   First of all this is the case that matters. Second the "closed" very badly collides with the notion of closed category.

   But, yes, we should at least add a remark on terminology. I don't have the time. Would very much appreciate if you add something.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 25th 2010
   • (edited Mar 25th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownI will do that, but I'm just not sure that my statement about the model structure on Top is correct. Edit: I know for a fact that the model structure on CGWH is closed, though.

   I will do that, but I'm just not sure that my statement about the model structure on Top is correct.

   Edit: I know for a fact that the model structure on CGWH is closed, though.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 25th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownQuillen's original word "closed" has been dropped by basically all workers in model category theory nowadays, having realized that "non-closed" model categories are not very common or useful. I don't know what you mean about topological spaces; Quillen's "closedness" condition was about cofibrations and fibrations determining each other by lifting properties; I don't think it says anything about homotopy equivalences.

   Quillen's original word "closed" has been dropped by basically all workers in model category theory nowadays, having realized that "non-closed" model categories are not very common or useful. I don't know what you mean about topological spaces; Quillen's "closedness" condition was about cofibrations and fibrations determining each other by lifting properties; I don't think it says anything about homotopy equivalences.

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 25th 2010
   • (edited Mar 25th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownI could have sworn that I read in one of the 6 sources I looked through that closedness => weak equivalences are homotopy equivalences, but I may have misread it (this is probable, since I was not reading the sources, but rather "finding in page" and scrolling through the results).

   I could have sworn that I read in one of the 6 sources I looked through that closedness => weak equivalences are homotopy equivalences, but I may have misread it (this is probable, since I was not reading the sources, but rather "finding in page" and scrolling through the results).