16
votes
3answers
333 views
What are surprising examples of Model Categories?
Background
Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together w …
10
votes
3answers
346 views
How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?
The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually alwa …
3
votes
1answer
101 views
A Model Structure on Symmetric Monoidal Categories
The recent article found here revisits Thomason's proof that symmetric monoidal categories model all connective spectra, but stops short of showing that there is a full closed mode …
1
vote
1answer
132 views
Analogs of left, right, inner, and Kan fibrations in CGWH
It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. Howe …
13
votes
5answers
511 views
Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-ske …
9
votes
2answers
205 views
Homotopy Limits over Fibered Categories
Suppose I have a small category $ \mathcal{C} $ which is fibered over some category $\mathcal{I}$ in the categorical sense. That is, there is a functor $\pi : \mathcal{C} \rightar …
13
votes
3answers
284 views
Is there an additive model of the stable homotopy category?
Is there a model category $C$ on an additive category such that its homotopy category $Ho(C)$ is the stable homotopy category of spectra and the additive structure on $Ho(C)$ is in …
13
votes
2answers
436 views
Why do my quantum group books avoid homotopical language?
I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes c …
4
votes
2answers
226 views
Group completion theorem
Let $M$ be a topological monoid. How does the homology-formulation of the group completion theorem, namely (see McDuff, Segal: Homology FIbrations and the "Group-Completion" Theore …
4
votes
1answer
226 views
Serre spectral sequence with spectra
A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason …
8
votes
1answer
161 views
homotopy associative $H$-space and $coH$-space
Let $[X, Y]_0$ denote base point preserving homotopy classes of maps $X\rightarrow Y$. A multiplication on a pointed space $Y$ is a map $\phi: Y\times Y\rightarrow Y.$ From this ma …
8
votes
0answers
126 views
Complex orientations on homotopy
I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement abou …
6
votes
2answers
278 views
Looking for good conference this summer for homotopy theory.
I'm a 2nd year grad student and I'm looking for conferences/summer schools to attend this summer. I checked out the AMS calendar but couldn't find anything I found relevant there. …
4
votes
2answers
322 views
$(\infty,1)$-categories and model categories
I read several times that $(\infty,1)$-categories (weak Kan complexes, special simplicial sets) are a generalization of the concept of model categories. What does this mean? Can on …
5
votes
1answer
113 views
BU with tensor product H-space structure
Hi,
I came across the space $BU_\otimes$ when struggling with twisted K-theory. Segal proved that this is an H-space, right? I have read a dozen times by now that the group $[X, …
