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 ...
Dmitri Pavlov's user avatar
10 votes

Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?

In the ten years since this question was asked, there has been a lot of progress in algebraic $K$-theory. For example, Achim Krause, Ben Antieau, and Thomas Nikolaus came up with an algorithm to ...
David White's user avatar
  • 29.4k
9 votes
Accepted

Which spectra have a universal connective quotient?

This answer is about the $\infty$-categorical variant. This is a fancy way to say: on spaces of maps, the natural map $$ Map(T',A) \to Map(T,A) $$ is an equivalence for any connective $A$. Note that ...
Tyler Lawson's user avatar
  • 51.1k
8 votes
Accepted

Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?

This seems to be an answer, based on discussion with Maxime Ramzi in the comments. The space of $E_\infty$ $KU$-algebra maps from $KU\otimes S^1_+\to F(S^1_+,KU)$ is the same as the space of $E_\infty$...
Neil Strickland's user avatar
7 votes

The complex $K$-theory of the Thom spectrum $MU$

You can learn about this, and more, in Part II of Adams' "Stable Homotopy and Generalised Homology" (1974), especially section 4, in the special case $E = KU$.
John Rognes's user avatar
  • 8,692
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\...
Dmitri Pavlov's user avatar
5 votes
Accepted

The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

Let $R$ be a ring. $BGL(R)^+$ is homotopy equivalent to the $0$ component of $\Omega^\infty K(R)$, and it is stably equivalent to $BGL(R)$. In particular, for a (co)homology theory $E$, understanding $...
Maxime Ramzi's user avatar
  • 13.3k
5 votes
Accepted

Stabilizing conjugacy classes of integer matrices

${\rm Conj}(A)\to{\rm Conj}(A\oplus I_m)$ is not surjective, because $$I_m\oplus A\in{\rm Conj}(A\oplus I_m) \quad\hbox{but}\quad\not\in R({\rm Conj}(A)).$$
Denis Serre's user avatar
  • 51.5k
5 votes

What are surprising examples of Model Categories?

There is Krause and Nikolaus' model structure on the category of group presentations whose homotopy category is the category of groups.
Tim Campion's user avatar
  • 60.6k
5 votes

Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?

No, this is not true, and nothing like this is expected. The exact sequence $\Bbb Z/2 \to \Bbb Z/4 \to \Bbb Z/2$ gives rise to a fiber sequence $H\Bbb Z/2 \to H\Bbb Z/4 \to H\Bbb Z/2$ of Eilenberg--...
Tyler Lawson's user avatar
  • 51.1k
4 votes
Accepted

Cohomology classes coming from algebraic K-theory

Apologies for being a bit late, but let me try to expand on my comment. An excellent source for this material is James Lewis' "user-friendly" survey article [Lew14]. First recall that for $X$...
Oli Gregory's user avatar
  • 1,259
4 votes

Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

The following is relatively standard. It is almost certainly not the "deep" reason you are asking for. I think it is useful point of view though. Suppose I have some invariant $J$ which ...
Geordie Williamson's user avatar
4 votes
Accepted

$K_0$ group of an infinite factor

The following is a more streamlined version of the answer I provided in the comments: We first observe that an infinite factor is properly infinite. (This is because a von Neumann algebra is defined ...
David Gao's user avatar
  • 1,262
3 votes

The complex $K$-theory of the Thom spectrum $MU$

For a more self-contained answer, let $L$ be the tautological line bundle over $\mathbb{C}P^\infty$. This gives a class $x=[L]-[1]\in\widetilde{K}^0(\mathbb{C}P^\infty)$. It is standard that $K^0(\...
Neil Strickland's user avatar
3 votes
Accepted

Equivariant sheaves on $\mathbb P^1$

Let me explain why the line bundle $\mathcal{O}(1)$ does not admit a $\mathrm{PGL}(2)$-equivariant structure. Indeed, if it does, then the vector space $$ \mathrm{Hom}(\mathcal{O}, \mathcal{O}(1)) $$ ...
Sasha's user avatar
  • 37k
3 votes
Accepted

"High-dimensional" classes in topological $K$-theory

Let $X_n = S^{2n+2}$. Since $\operatorname{ch} : K(S^{2n+2})\otimes_{\mathbb{Z}}\mathbb{Q} \to H^{\text{even}}(S^{2n+2}; \mathbb{Q})$ is an isomorphism, there is a complex vector bundle $E \to S^{2n+2}...
Michael Albanese's user avatar
3 votes
Accepted

$K_1$ of Categories for intuition

For an exact category ${\cal N}$, Dan Grayson (who occasionally shows up on MathOverflow) gave explicit generators and relations for $K_n$ here. What follows is essentially quoted directly from that ...
Steven Landsburg's user avatar
3 votes
Accepted

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

This sounds like a very exciting phase of your studies! As you point out, there's plenty of intersections between algebraic geometry and homotopy theory. One intersection is motivic homotopy theory, ...
David White's user avatar
  • 29.4k
2 votes

Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

I certainly respect going back to primary sources. But, in this case, it's helpful to remember that a LOT has been written about bisimplicial sets since Quillen's 1973 paper. For example, a reference ...
David White's user avatar
  • 29.4k
1 vote

Application of higher categories in algebra

There are tons of such applications. For example, less than two years ago, they were used to prove the Redshift Conjecture, a statement about iterated algebraic $K$-theory. I previously wrote an ...
1 vote

Does Grayson/Quillen's "pre group completion" have a universal property?

It is the classifying category for the left action of $C$ on its product $C \times C$. Let me elaborate on this a bit further. Let $C$ be an $E_\infty$-monoid, for example represented by a symmetric ...
Georg Lehner's user avatar
  • 1,943
1 vote

The category theory of Span-enriched categories / 2-Segal spaces

A solution to achieving the kind of enrichment you are looking for (though I cannot say what precise comparison it would give with $2$-Segal spaces) may be to actually remember not only the $2$-...
David Kern's user avatar
1 vote

Integral group rings on which stably free modules are free

This is an old question but I thought this could be a useful answer to anyone interested in this in the future. The only class of examples of torsion-free groups with SFC (stably-free cancellation) ...
William Thomas's user avatar

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