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
   • CommentTimeDec 28th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexgiving this its own entry (the concept used to appear in-line at *[[geometric realization of simplicial topological spaces]]*) for ease of hyperlinking. But just the bare definition, for the moment. <a href="https://ncatlab.org/nlab/revision/homotopy+Kan+fibration/1">v1</a>, <a href="https://ncatlab.org/nlab/show/homotopy+Kan+fibration">current</a>

   giving this its own entry (the concept used to appear in-line at geometric realization of simplicial topological spaces) for ease of hyperlinking. But just the bare definition, for the moment.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeApr 23rd 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded another link in Related concepts section anonymous <a href="https://ncatlab.org/nlab/history/show/homotopy+Kan+fibration/2">v2</a>, <a href="https://ncatlab.org/nlab/show/homotopy+Kan+fibration">current</a>

   added another link in Related concepts section

   anonymous

   v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 4th 2022
   • (edited Jul 4th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am suspecting that the following is true, but I don't have a proof yet. Possibly this is an easy consequence of statements that are in the literature. Given 1. a morphism $X_\bullet \overset{p_\bullet}{\longrightarrow} B_\bullet$ of simplicial k-topological spaces which is a homotopy Kan fibration 1. a Čech nerve $C(U)_\bullet$ of a good open cover over a manifold, also regarded as a simplicial topological space, and writing * $Map \,:\, sTop^{op} \times sTop \longrightarrow sTop$ for the simplicial topological mapping complex between simplicial k-topological spaces then presumably * $Map\big( C(U)_\bullet, \, X_\bullet \big) \overset{Map\big(C(U)_\bullet, p_\bullet\big)}{\longrightarrow} Map\big( C(U)_\bullet , B_\bullet\big)$ is again a homotopy Kan fibration?! (What I am really after is that its realization commutes with taking homotopy fibers.) Presumably I'll have to convince myself that $C(U)_\bullet$ is "homotopy cofibrant" in the suitable sense and that simplicial $\infty$-groupoids, regarded as a "model $\infty$-category" is "cartesian closed model". All this sounds like it ought to be true. Can this be cited from anywhere?

   I am suspecting that the following is true, but I don’t have a proof yet. Possibly this is an easy consequence of statements that are in the literature.

   Given

   1. a morphism $X_\bullet \overset{p_\bullet}{\longrightarrow} B_\bullet$ of simplicial k-topological spaces which is a homotopy Kan fibration

   2. a Čech nerve $C(U)_\bullet$ of a good open cover over a manifold, also regarded as a simplicial topological space,

   and writing

   • $Map \,:\, sTop^{op} \times sTop \longrightarrow sTop$ for the simplicial topological mapping complex between simplicial k-topological spaces

   then presumably

   • $Map\big( C(U)_\bullet, \, X_\bullet \big) \overset{Map\big(C(U)_\bullet, p_\bullet\big)}{\longrightarrow} Map\big( C(U)_\bullet , B_\bullet\big)$

   is again a homotopy Kan fibration?! (What I am really after is that its realization commutes with taking homotopy fibers.)

   Presumably I’ll have to convince myself that $C(U)_\bullet$ is “homotopy cofibrant” in the suitable sense and that simplicial $\infty$-groupoids, regarded as a “model $\infty$-category” is “cartesian closed model”.

   All this sounds like it ought to be true. Can this be cited from anywhere?

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2022
   • (edited Jul 6th 2022)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexThe Čech nerve C(U) of a good open cover over a manifold, regarded as a simplicial topological space, is weakly equivalent to a discrete space in every simplicial degree. Such simplicial spaces are cofibrant, since they can be obtained as transfinite compositions of cobase changes of generating cofibrations in the model ∞-structure on simplicial spaces, namely, boundary inclusions ∂Δ^n→Δ^n. This answers the question in the affirmative. I guess one could cite Mazel-Gee's articles, Part I deals specifically with simplicial spaces.

   The Čech nerve C(U) of a good open cover over a manifold, regarded as a simplicial topological space, is weakly equivalent to a discrete space in every simplicial degree.

   Such simplicial spaces are cofibrant, since they can be obtained as transfinite compositions of cobase changes of generating cofibrations in the model ∞-structure on simplicial spaces, namely, boundary inclusions ∂Δ^n→Δ^n.

   This answers the question in the affirmative. I guess one could cite Mazel-Gee’s articles, Part I deals specifically with simplicial spaces.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 6th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexBut how about the property that homming a homotopy-cofibrant simplicial object into a homotopy Kan fibration yields a homotopy Kan fibration (or more generally the $\infty$-version of the [[pullback-power axiom]]). Is that proven anywhere?

   But how about the property that homming a homotopy-cofibrant simplicial object into a homotopy Kan fibration yields a homotopy Kan fibration (or more generally the $\infty$-version of the pullback-power axiom). Is that proven anywhere?

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexYes, this follows from Proposition 5.6 in Mazel-Gee II, which shows that a derived two-variable Quillen adjunction induces a closed symmetric monoidal structure on the localization. Since the monoidal product of two terminal objects is the terminal object, this resulting monoidal structure is cartesian. Thus, the hom from an ∞-cofibrant to an ∞-fibrant simplicial space computes the correct derived hom with respect to diagonal weak equivalences.

   Yes, this follows from Proposition 5.6 in Mazel-Gee II, which shows that a derived two-variable Quillen adjunction induces a closed symmetric monoidal structure on the localization.

   Since the monoidal product of two terminal objects is the terminal object, this resulting monoidal structure is cartesian.

   Thus, the hom from an ∞-cofibrant to an ∞-fibrant simplicial space computes the correct derived hom with respect to diagonal weak equivalences.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 6th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the pointer! Had not looked at part II before. I'll need to think about how that Prop. 5.6 implies the claim. At face value it looks like claiming something slightly weaker, no? But i see that the pushout-product axiom is in Def. 5.1, and it holds for simplicial spaces by Ex. 5.3. Now if that example 5.3 simply included the word "closed", we'd immediately have the pullback-power axiom by the usual dualization.

   Thanks for the pointer! Had not looked at part II before.

   I’ll need to think about how that Prop. 5.6 implies the claim. At face value it looks like claiming something slightly weaker, no?

   But i see that the pushout-product axiom is in Def. 5.1, and it holds for simplicial spaces by Ex. 5.3. Now if that example 5.3 simply included the word “closed”, we’d immediately have the pullback-power axiom by the usual dualization.

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexConcerning Example 5.3: it is closed by Definition 4.3, and Section 4 does establish the pullback-power axiom.

   Concerning Example 5.3: it is closed by Definition 4.3, and Section 4 does establish the pullback-power axiom.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJul 6th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, I see, it's built into Def. 4.3. Great, thanks again.

   Oh, I see, it’s built into Def. 4.3. Great, thanks again.