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    • We had (still have) a proof of the contractibility of some version of S ∞S^\infty in the Definition-section at n-sphere.

      Since that doesn’t seem to be the right place for that material, and in order to make it easier to link to S ∞S^\infty and its contractibility, I am giving it its own page here.

      In creating this page, I have:

      • copied over the material of the section n-sphere – Definition – Infinite-dimensional sphere;

      • expanded out the first paragraph into a new Idea-section here;

      • added a section with the definition as a colimit over relative cell complex inclusions and the quick proof of weak contractibility from that.

      So the previous discussion in terms of infinite-dimensional unit spheres in LCTVSs and/or in shift spaces is currently both here as well as inside n-sphere. But I suggest we remove it at the latter place, and just leave the link to this new page here.

      v1, current

    • I am giving this its own page, in order to have a way to point to quaternionic conjugation etc.

      (Previously “complex conjugation” just redirected to “complex number”.)

      v1, current

    • I have given complex conjugation its own page, in order to have a way to point to quaternionic conjugation etc. (Previously “complex conjugation” just redirected to “complex number”.)

      But the system has a hiccup: The page exists now, but the announcement didn’t get through to here. And any attempt to edit the page first leads to the system claiming that I have locked the page and, ignoring that, to a 500 error message.

      So I can’t fix the page now. I’ll leave it as is for the time being.

    • The first line of the Idea-section read:

      Every magma AA has an opposite A opA^op in which the operation goes the other direction.

      This rather sounded like talking about co-magmas. I have replaced this now with the following more lengthy but less ambiguous sentence:

      The opposite of a magma – hence of a set with a binary operation (x,y)↦xy(x,y) \mapsto x y – has the same underlying set of elements, but binary operation changed by reversing the order of the factors: (x,y)↦yx(x,y) \mapsto y x.

      Also I touched the Definition-section, trying to beautify a little, both the wording and the formulas.

      diff, v4, current

    • create page for prime knot. WIP..

      Grant

      v1, current

    • A stand-alone page, with a detailed proof/explanation for the lemma that B𝕂 ×→0Th(ℒ B𝕂 × *)B \mathbb{K}^\times \overset{0}{\to} Th(\mathcal{L}^\ast_{B\mathbb{K}^\times}) is a weak homotopy equialence.

      This was (and is) stated with the traditional sketchy proof on the page universal complex orientation on MU. This stand-alone page here since it’s awkward to point to within that page just for this lemma, and also in order to generalize beyond the complex ground field and to have more room to talk about the proof.

      v1, current

    • fixed a typo

      (the complex numbers appeared as “𝔹\mathbb{B}” a few lines in a row, apparently copy-and-pasted onwards)

      diff, v17, current

    • In articles by Balmer I see “tensor monoidal category” to be explained as a triangulated category equipped with a symmetric monoidal structure such that tensor product with any object “is an exact functor”, but I don’t see where he is specific about what “exact functor” is meant to mean. Maybe I am just not looking in the right article.

      Clearly one wants it to mean “preserving exact triangles” in some evident sense. One place where this is made precise is in def. A.2.1 (p.106) of Hovey-Palmieri-Strickland’s “Axiomatic stable homotopy theory” (pdf).

      However, these authors do not use the terminology “tensor triangulated” but say “symmetric monoidal compatible with the triangulation”. On the other hand, Balmer cites them as a reference for “tensor triangulated categories” (e.g. page 2 of his “The spectrum of prime ideals in tensor triangulated categories” ).

      My question is: may I assume that “tensor triangulated category” is used synonymously with Hovey-Palmieri-Strickland’s “symmetric monoidal comaptible with the triangulation”?

    • Preliminary notes for symanzik effective theory in the context of scale setting procedures for calculating hadronic observables using chiral perturbation theory, which involves discretization corrections, where the SET is employed.

      Grant

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • I have type the definition of multiplicative unreduced generalized cohomology theories AA into multiplicative cohomology theory. Then I added the statements and their arguments (here) for the compatible A •(*)A^\bullet(\ast)-module structure on AA-cohomology groups and the A •(*)A^\bullet(\ast)-linearity of the differentials in any Atiyah-Hirzebruch spectral sequence with coefficients in AA.

    • Seeing this old entry again, I have touched the wording, formatting, organization and hyperlinking .

      diff, v13, current

    • Page created, but author did not leave any comments.

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • I have touched formal group a bit, but don’t have time to do anything substantial.

      I need to adjust some of the terminology that I had been setting up at cohesive (infinity,1)-topos related to infinitesimal cohesion : the abstract notion currently called “∞\infty-Lie algebroid” there should be called “formal cohesive ∞\infty-groupoid”. The actual L-infinity algebroids are (just) the first order formal smooth ∞\infty-groupoids.

      While on the train I started expanding some other entries on this point, but I need to quit now and continue after a little interruption.

    • starting something – not done yet, but need to save

      v1, current

    • Changed “interval functor” to “cylinder functor” in “While this is the motivating example, the interval functor need not be of that form”, in Remarks.

      diff, v10, current

    • Page created, but author did not leave any comments.

      eldrago

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • expanded (here) the remark on homotopy groups of Omega-spectra just a bit more

      diff, v7, current

    • noticed that the Idea-section at ring spectrum didn’t at all address the evident subtlety here. Have expanded now to make this clear.

    • I have added pointer to the second of Postnikov’s original articles on the matter:

      • M. M. Postnikov, Issledovaniya po gomotopičeskoĭ teorii nepreryvnyh otobraženiĭ. I. Algebraičeskaya teoriya sistem. II. Naturalʹnaya sistema i gomotopičeskiĭ tip. (Russian) [[_Investigations in homotopy theory of continuous mappings. I. The algebraic theory of systems. II. The natural system and homotopy type._]] Trudy Mat. Inst. Steklov. no. 46. Izdat. Akad. Nauk SSSR, Moscow, 1955. (mathnet:tm1182)

      Is there any linkable online trace of Postnikov’s first article:

      • M. M. Postnikov, Determination of the homology groups of a space by means of the homotopy invariants, Doklady Akad. Nauk SSSR (N.S.) 76: 359–362 (1951)

      ?

      diff, v53, current

    • fixed notation in the second formula in the proof of this Prop.:

      (The adjoined base point (−) +(-)_+ to the symmetric group factor was previously displayed below the formula beneath the underbrace below the symmetric group symbol that it really belonged to. And in the second line of that formula under the brace, the corresponding (−) +(-)_+ had been missing.)

      diff, v56, current

    • coming here to add “selected writings”, I have adjusted/updated the wording a little

      diff, v17, current

    • have expanded the list of “Selected writings”: more entries, more complete bib-info

      diff, v3, current

    • have been expanding the list of examples slightly

      diff, v6, current

    • starting something, on Conner-Floyd’s (U,fr)(U,fr)-bordism theory

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • starting something, but am being interrupted now. Am saving anyway to make the link work

      v1, current

    • I added a note about what “projectively flat” means in the context of parabolic Cartan geometry. I would assume that this should mean the same thing as “a connection which is flat up to central term” as described in the first sentence?

      diff, v2, current

    • That doesn’t look right at Via left homotopy of spectra. ℓ\ell is supposed to be the forgetful functor from spectra to prespectra.

      But what kind of spectra are we looking at? It seems to be coordinate-free spectrum. So then we need a definition of prespectrum.

    • Earlier today I was checking where on the nnLab we had recorded basics on finite homotopy (co)limits of spectra. But it seems we haven’t at all, except for the discussion at Introduction to Stable homotopy theory.

      So then I started to add something at Spectra, only to notice that this needs harmonizing/merging with the parallel entry stable (infinity,1)-category of spectra.

      To cut this Gordian knot, I am now creating hereby an entry with a bare section on finite homotopy (co)limits of spectra, to be !includeed into these entries (and into stable homotopy category and maybe elsewhere, too).

      So far I have just some bare minimum here. Deserves to be expanded.

      v1, current

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