added at cobordism hypothesis a pointer to

- Yonatan Harpaz,
*The Cobordism Hypothesis in Dimension 1*(arXiv:1210.0229)

where the case for $(\infty,1)$-categories is spelled out and proven in detail.

]]>I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.

For the record, I found the inductive characterisation in Cisinski’s book [*Les préfaisceaux comme modèles des types d’homotopie*, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [*Notes on simplicial homotopy theory*], but they take it as a *definition*.

added to [[group cohomology]]

in the section structured group cohomology some remarks about how to correctly define Lie group cohomology and topological group cohomology etc. and how not to

in the section Lie group cohiomology a derivation of how from the right oo-categorical definition one finds after some unwinding the correct definition as given in the article by Brylinski cited there.

it's late here and I am now in a bit of a hurry to call it quits, so the proof I give there may need a bit polishing. I'll take care of that later...

]]>I have added to *monoidal model category* statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

$\gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.$The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

]]>I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

]]>created a minimum at *real homotopy theory*

I have tried to give *algebraic topology* a better Idea-section.

stub for *Blakers-Massey theorem*. Need to add more references…

I have created *Sullivan model of free loop space* with the formula and pointers to the literature.

I need a word for the homotopy quotient $(\mathcal{L}X)/S^1$ of free loop spaces $\mathcal{L}X$ by their canonical circle action. It seems that the only term in use with respect to this is “twisted loop space”, which however usually refers just to the constant loops $(\mathcal{L}_{const}X)//S^1$. Since under nice conditions the derived functions on the $\mathcal{L}Spec(A)/S^1$ is the cyclic homology complex of $A$, I suggest that a good name is “cyclic loop space”. I made a quick note at *cyclic loop space*, just to fix and disambiguate terminology.

created [[motive]] just in order to link to the sub-pages on this that we already have, and in order to record a link to a useful MO discussion about them.

]]>I have expanded, streamlined and re-organized a little at *differential forms on simplices*.

created *red-shift conjecture*

… whereby the periodic table is fianlly un-grayed

]]>had created *braided 3-group* a good while back. Now I have added the example of Brauer/Picard/Unit-3-groups and cross-linked with *Brauer group*.

I added to *cylinder object* a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex).

What would be a text that features a *graphics* which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

in reaction to an email discussion I had, I have finally added to the section *Derived hom-spaces* at *category of fibrant objects* the definition and theorem that had been alluded to there all along.

I have added reference to Gabriel-Zisman’s *Calculus of fractions and homotopy theory* to the following entries:

and maybe others. Usually, I’d conversely create a hyperlinked index at *Calculus of fractions and homotopy theory*, but no time right now.

brief entry *model structure on semi-simplicial sets*, just in order to record a recent note by Benno van den Berg.

have split off *semi-simplicial object* from *semi-simplicial set*.

added to *homotopy fixed point* a discussion of how the traditional ad-hoc formula that one finds in much of the literature (namely $X^{h G} = Hom_G(E G, X)$) follows form first principles.

(This is for completeness, not because it is a big deal.)

]]>At the old entry *cohomotopy* used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

]]>I had given it an $n$Lab page already a while back, so that I could stably link to it without it being already there:

Now it’s even “there” in the sense of being incarnated as a pdf.

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