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.
   • CommentAuthorUrs
   • CommentTimeMar 25th 2010
   • PermaLink
   Author: Urs
   Format: Markdownstub for [[model structure on algebraic fibrant objects]] just the bare minimum for the moment, no time...

   stub for model structure on algebraic fibrant objects

   just the bare minimum for the moment, no time...

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexexpanded the entry [[model structure on algebraic fibrant objects]]: added an Idea-section and an Applications-section. Also created stub entries * [[algebraic Kan complex]] * [[algebraic quasi-category]] and tried to add all the relevnt cross-links to and from other relevant entries.

   expanded the entry model structure on algebraic fibrant objects: added an Idea-section and an Applications-section.

   Also created stub entries

   • algebraic Kan complex

   • algebraic quasi-category

   and tried to add all the relevnt cross-links to and from other relevant entries.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexand interlinked with [[simplicial T-complex]]

   and interlinked with simplicial T-complex

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI wrote out at [[model structure on algebraic fibrant objects]] all the details for the proof of the monadicity and the model structure, or rather all the details establishing monadicity and solidity of the forgetful functor, then pointing over to [[solid functor]] for the conclusion.

   I wrote out at model structure on algebraic fibrant objects all the details for the proof of the monadicity and the model structure, or rather all the details establishing monadicity and solidity of the forgetful functor, then pointing over to solid functor for the conclusion.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI added further details. Effectively, I have now spelled out the full proof and all the constructions that go into it.

   I added further details. Effectively, I have now spelled out the full proof and all the constructions that go into it.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement and proof that $Alg C$ is locally presentable/combinatorial if $C$ is -- <a href="http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects#LocalPresentability">here</a>

   added statement and proof that $Alg C$ is locally presentable/combinatorial if $C$ is – here

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 17th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think there is a terminological problem here, which I'm sorry I didn't catch back when it was getting written on the page. A [[solid functor]] has semi-final lifts for *all* structured sinks, not just small ones; but as far as I can see, in general algebraic fibrant objects only have semi-final lifts for *small* structured sinks. (Though it seems possible that some kind of well-copoweredness condition might make them actually solid.) I don't think I've heard of a name for functors with this property, but we could invent one since it seems a natural condition. Quasi-solid? Small-solid? Plastic? [Supersolid](https://en.wikipedia.org/wiki/Supersolid)? By the way, I just learned that [this paper](https://arxiv.org/abs/1712.02523) of John Bourke proves Nikolaus's result for any combinatorial model category without the assumption that acyclic cofibrations are monic. Instead of the semi-final lifts he uses the path object argument to prove acyclicity. We should add this to the page too, although certainly we should keep the explicit description of Nikolaus's method too since it has other applications.

   I think there is a terminological problem here, which I’m sorry I didn’t catch back when it was getting written on the page. A solid functor has semi-final lifts for all structured sinks, not just small ones; but as far as I can see, in general algebraic fibrant objects only have semi-final lifts for small structured sinks. (Though it seems possible that some kind of well-copoweredness condition might make them actually solid.) I don’t think I’ve heard of a name for functors with this property, but we could invent one since it seems a natural condition. Quasi-solid? Small-solid? Plastic? Supersolid?

   By the way, I just learned that this paper of John Bourke proves Nikolaus’s result for any combinatorial model category without the assumption that acyclic cofibrations are monic. Instead of the semi-final lifts he uses the path object argument to prove acyclicity. We should add this to the page too, although certainly we should keep the explicit description of Nikolaus’s method too since it has other applications.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 17th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThere's also some abstract discussion of such semi-final lifts for categories of "algebraic injectives" in section 4 of [this paper](https://arxiv.org/abs/1811.09532) (also by Bourke). Maybe it would be better to move some of the results that work abstractly to a new page like [[algebraic injective]].

   There’s also some abstract discussion of such semi-final lifts for categories of “algebraic injectives” in section 4 of this paper (also by Bourke). Maybe it would be better to move some of the results that work abstractly to a new page like algebraic injective.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 18th 2019
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #7: I like small-solid best, as the most descriptive.

   Re #7: I like small-solid best, as the most descriptive.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 18th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHmm... actually, now that I was reminded that all locally presentable categories are well-copowered, perhaps we do get full solidity in that case. Suppose we have a (possibly large) $U$-structured sink $\{U Y_i \to X\}$. Each of those maps induces its own map $X \to X_{H_i}$, which is an epimorphism. In general there might be a large number of these, but in a well-copowered category we can take cointersections of even large families of epimorphisms to obtain a single epimorphism $X\to X_{H}$, which ought to be what we need to make the rest of the proof go through.

    Hmm… actually, now that I was reminded that all locally presentable categories are well-copowered, perhaps we do get full solidity in that case. Suppose we have a (possibly large) $U$-structured sink $\{U Y_i \to X\}$. Each of those maps induces its own map $X \to X_{H_i}$, which is an epimorphism. In general there might be a large number of these, but in a well-copowered category we can take cointersections of even large families of epimorphisms to obtain a single epimorphism $X\to X_{H}$, which ought to be what we need to make the rest of the proof go through.