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Results tagged with kt.k-theory-and-homology
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user 402
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
12
votes
What is decategorification?
Taking loops (or, in categorical language, endomorphisms of the monoidal unit) is commonly seen as a type of decategorification.
For example, decategorifying von Neumann algebras produces Hilbert spac …
- 37.8k
6
votes
Accepted
Can one bypass the geometric realization in the definition of algebraic $K$-theory?
I believe there is no good notion of homotopy groups for an arbitrary simplicial set S.
It depends on what “good” means.
Kan's original definition works for arbitrary pointed simplicial sets: $$\def …
- 37.8k
1
vote
A question on the ring structure of topological K-theory and Chern character
The product of $K^{−1}$ with itself can be defined by identifying $K^{−1}(X)$
with $K^0(X\wedge S^1)$ using the suspension isomorphism, multiplying the resulting two elements of $K^0(X\wedge S^1)$, o …
- 37.8k
7
votes
Invertibility of element in $K(X)$
Yes. Any rank zero element x in K(X) is nilpotent by https://ncatlab.org/nlab/show/virtual%20vector%20bundle, hence 1+x is invertible.
- 37.8k
3
votes
Accepted
Reference for $E_{\infty}$-ness of the Chern Character
The answer really depends on one's desired choice of definitions for KU, HQ, and the Chern character itself;
some definitions allow one to produce a very short definition of the Chern character as an …
- 37.8k
4
votes
K-theory of non-compact spaces
What exactly does 3. describe? Are these virtual vector bundles that admit numerable trivializations?
Virtual vector bundles, when defined as formal differences (i.e.,
elements in the homotopy gr …
- 37.8k
51
votes
Accepted
What are surprising examples of Model Categories?
Here is an example that surprised me at some time in the past.
Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs.
Here a directed graph is simply a $4$-tuple $(V, …
- 37.8k