created *dg-nerve*

added to homotopy groups of spheres the table

$k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\pi_k(\mathbb{S}) =$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | $0$ | $0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $(\mathbb{Z}_2)^2$ | $(\mathbb{Z}_2)^3$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | $0$ | $\mathbb{Z}_3$ | $(\mathbb{Z}_2)^2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ |

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

some basics at *Steenrod algebra*

added the pointers to the combinatorial proofs of the fiberwise detection of acyclicity of Kan fibrations, currently discussed on the AlgTop list, to the nLab here.

]]>created *fibrant type* with an Idea-section

I have added some basic content to the old entry *fibrant object*: An idea-section, the statement about (co)fibrant replacement, and a few basic examples.

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 have touched *H-space*, slightly expanded here and there and slightly reorganized it.

also created *axiom UIP*, just for completeness. But the entry still needs some reference or else some further details.

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

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 need to point to *[[reduced homology]]*, so I created an entry. But nothing much there yet.

stub for homotopy type theory

]]>added to the references-section of the stub *type-theoretic model category* pointers to André Joyal’s slides on “typoses” (he is currently speaking about that again at CRM in Barcelona).

(maybe that entry should be renamed to “categorical semantics for homotopy type theory” or the like, but I won’t further play with it for the time being).

I am also pointing to Mike’s article and to his course notes. I will maybe ask André later, but I am a bit confused about (was already in Halifax) how he presents his typoses, without mentioning of at least very similar categorical semantics that has been discussed before. Maybe I am missing some sociological subtleties here.

]]>following public demand, I added to *tensor product of chain complexes* a detailed elementary discussion of the tensor product $I_\bullet \otimes I_\bullet$ of the (normalized) chain interval with itself, and how it gives chains on the cellular square: in *Square as tensor product of interval with itself*.

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

started a minimum at *Bousfield-Friedlander theorem*

(the model category theoretic incarnation of idempotent $\infty$-monads)

cross-linked with brief paragraphs to *Bousfield-Friedlander model structure*

added to *polynomial monad* the article by Batanin-Berger on homotopy theory of algebras over polynomial monads.

I have created an entry on the *quaternionic Hopf fibration* and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in $G$-equivariant stable homotopy theory it represents a non-torsion element in

for $G$ a finite and non-cyclic subgroup of $SO(3)$, and $SO(3)$ acting on the quaternionic Hopf fibration via automorphisms of the quaternions.

I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.

]]>When I was about to create it for *flat connection* I notice that we already did have *Riemann-Hilbert correspondence*. So now I have cross-linked it with *flat connection*, *flat infinity-connection*, *local system*, *Riemann-Hilbert problem* and the latter with *Hilbert’s problems*

created Cisinski model structure

]]>I have added to all *Segal space*-related entries, as well as to the Example section at *category object in an (infinity,1)-category* statements like

a pre-category object in $\infty Grpd$ is called a Segal space;

a connected pre-category object in $\infty Grpd$ is called a reduced Segal space;

a category object in $\infty Grpd$ is called a complete Segal space.

an category object in $Cat(Cat(\cdots Cat(\infty Grpd)))$ is called an n-fold complete Segal space;

That list can be further expanded. But I have to quit now.

]]>I have touched the formatting at *free groupoid*. Then I added the statement that the fundamental groups of a free groupoid are free. Also added a pointer to a writeup of the proof.