Not signed in (Sign In)

Start a new discussion

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics 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 finite foundations functional-analysis functor galois-theory 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 lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number 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 string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 23rd 2019

    Gave proper reference for (Kieboom 1987).

    diff, v26, current

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeOct 11th 2020

    Fixed typo.

    diff, v28, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2021
    • (edited Sep 14th 2021)

    gave a reference to a proof for the claim that Hurewicz cofibrant closed subspace inclusions are equivalently those whose pushout-product with {0}[0,1]\{0\} \hookrightarrow [0,1] has a retraction

    and added (here) the observation that this implies, for all spaces involved being compactly generated, that k-ified products with a space preserves the Hurewicz cofibration property of subspace inclusions.

    diff, v30, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2021

    I have removed the previous text and diagrams in the Definition-section (here), whose organization and typesetting was awkward to the point of being unreadable, and replaced it with the polished typesetting of the same material that I just added also to the entry homotopy extension property.

    diff, v32, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2021
    • (edited Sep 15th 2021)

    I have added a pointer (here) to p. 44 in May’s “Concise AlgTop” for the claim that Hurewicz cofibrations in (weakly) Hausdorff spaces are always closed.

    (But May doesn’t spell out the argument either…)

    diff, v33, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTime6 days ago

    For what it’s worth, I have added a little TikZ-diagram hosting that retraction (here)

    What’s an actual proof that Lie groups are well-pointed?

    One could draw a cute diagram here, showing how “the group grows a spike at its neutral element”. For S 1S^1 this still has a reasonable rendering. Maybe later.

    diff, v36, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTime5 days ago
    • (edited 5 days ago)

    added (here) pointer to a couple places where proof is spelled out that relative CW-complexes are h-cofibrations.

    diff, v37, current

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTime5 days ago

    re 6: Strøm’s first note on cofibrations (1966: 12) has a characterization of closed cofibrations (theorem 2) as the inclusions of those closed subspaces AXA \subseteq X for which there exists an open UXU \subseteq X which is deformable rel AUA\subseteq U to AA in XX and for which there is a function φ:X[0,1]\varphi\colon X\to [0,1] with φ 1(0)=A\varphi^{-1}(0)=A and φ 1(1)=XU\varphi^{-1}(1)=X\setminus U. That shows that every locally Euclidean Hausdorff space is well-pointed at all of its points. (For xXx\in X, take a chart UXU\subseteq X around xx, and an open ε\varepsilon-ball VUV\subseteq U around xx; there is a function φ:U[0,1]\varphi\colon U\to[0,1] with φ 1(0)=x\varphi^{-1}(0)=x and φ 1(1)=UV\varphi^{-1}(1)=U\setminus V, which can be extended to XX by taking it to be 11 on the complement of the closed (compact!) ε\varepsilon-ball in UU around xx.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTime4 days ago
    • (edited 4 days ago)

    Thanks for the hints!

    Have now added the content of Strøm’s Thm. 2 here, drawing some diagrams to make the zoo of conditions a little more transparent (to my mind, at least).

    Will next spell out in the entry the conclusion about Euclidean spaces that you indicate. Is there are citable reference that makes this conclusion explicit?

    diff, v39, current

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTime4 days ago
    I don’t know one off the top of my head (or after rummaging through my pdf folder for a bit); Bredon (1993: 432) also has Strøm’s theorem 2 (as theorem 1.5, and with a proof of continuity this time), but thinks the well-pointedness of pointed manifolds is obvious (bottom of p. 435, below definition 1.8). I guess that’s an all too common opinion…
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Thanks. Yes, I had already seen it in Bredon’s book and added the pointer. That seems to be a pretty good book, have only opened it tonight for the first time.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTime4 days ago

    It’s getting a bit late for me, maybe I need to call it quits and continue tomorrow. What I am really after is verifying that PU(ℋ) and friends are well-pointed. It should all be exercise-level checking, but it’s also a little annoying…

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTime4 days ago
    • (edited 4 days ago)

    Okay, I have now added the example of locally Euclidean spaces here.

    By the way, we are not required to have ϕ 1({1})=XU\phi^{-1}(\{1\}) \,=\, X\setminus U, just ϕ 1({1})XU\phi^{-1}(\{1\}) \,\supset\, X\setminus U, no? On the other hand, using the open ball as the neighbourhood we would have this equality. Either way it works, it seems to me.

    Or maybe I am too tired now, will call it a day.

    diff, v40, current

    • CommentRowNumber14.
    • CommentAuthorGuest
    • CommentTime4 days ago

    It’s late here too, so as a note to myself I’ll leave Wikipedia on ANRs and Aguilar–Gitler–Prieto (2002: 99)’s theorem 4.2.15 (a closed embedding AXA\embedsin X with XX an ANR is a closed cofibration iff AA is an ANR – which, they remark, includes paracompact Hausdorff manifolds modeled on Banach spaces, cf. p.99, just after exercise 4.2.11). BTW, you do need some kind of separation property on your locally Euclidean space to make it well-pointed at every point; for example the line with two origins is not well-pointed at either origin since real-valued functions can’t tell them apart (Hausdorffness or KC (=every compact set is closed) was a quick fix). I’ll sleep on the possibly strict inclusions :)

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Thanks for the pointer to AGP!, I’ll check that out now.

    Regarding separation. Thanks, I guess I forgot to check that extension of ϕ |U\phi_{\vert U} by the function constant on 1 be continuous. So what do you appeal to for this to be the case? Tietze?

    • CommentRowNumber16.
    • CommentAuthorGuest
    • CommentTime4 days ago

    Hausdorffness is enough (maybe even necessary): the closed ball of radius 11 is compact and hence closed in (Hausdorff) space as a whole, so its complement is open, and a function which is continuous on each set in an open cover is continuous. (I’ll try and edit that in, if the train wifi allows it)

  1. Added a continuity argument.

    Anonymous

    diff, v43, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Thanks!

    Just for my own benefit I have expanded out your argument a tad more (here).

    diff, v44, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Ah, I see there are typos left, but you are editing now.

  2. Looks good! Made the domain of the homotopy η\eta explicit, and fixed UU being used for two different things (you don’ need an extra ε>0\varepsilon \gt 0, BTW)

    Anonymous

    diff, v45, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Okay, maybe we are editing cross-purpose now. What is UU in your latest version, it seems to be undefined now? I thought with taking UU to be any open ball a little larger than the unit ball I was implementing your proof idea. In your latest version UU seems undefined. It seems to me if we change “VV” back to “UU” and declare it to be a little larger than unity, then we have a proof. No?

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Oh, sorry, I see now that you use UU for the chart. Could we then swap UVU \leftrightarrow V so that the notation matches that of the Proposition we are appealing to.

    • CommentRowNumber23.
    • CommentAuthorGuest
    • CommentTime4 days ago
    Good point, I forgot about that. I’ll leave the editing to you, then :)
    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Okay, thanks. I have just slightly edited, making the naming of the chart more explicit and taking the domain of η\eta to be U×[0,1]U \times [0,1]. In fact then it seems that we can just do away with VV and the proof runs nicely. Hope I am not overlooking something.

    (Sorry if I have been making it more complicated than necessary, thanks again for all your input!)

    Next, I have now typed out the argument, via a couple of references, that closed Banach submanifold inclusions into paracompact Banach manifolds are h-cofibrations: here.

    That’s great, this gives all the well-pointedness I need for dealing with the universal equivariant PU(H)-bundle. :-)

    diff, v46, current

    • CommentRowNumber25.
    • CommentAuthorGuest
    • CommentTime4 days ago

    Looks good to me (and you were right about merely needing an inclusion XvarsetminusUϕ 1(1)X\varsetminus U\subseteq \phi^{-1}(1) — I didn’t notice that that’s how Bredon has it). Sorry for interfering with your edits earlier! It’s great to have this stuff written up somewhere :)

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTime4 days ago

    My pleasure. Thanks again for your help.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTime4 days ago

    Oh no, I see now I was wrong to think that U(H) is Banach in the strong topology. It is so only in the norm topology. Need to work harder to show that in the norm-topology it’s still well-pointed…

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTime4 days ago

    added (here) mentioning of the example of the point inclusion into PU()PU(\mathcal{H})

    diff, v47, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTime2 days ago

    added (here) statements that

    • composition of h-cofibs is an h-cofib

    • inclusion into a normal space is h-cofib iff it is so with respect to any open neighbourhood

    diff, v50, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)