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Let me make some guesses, although I worry I'll only say some formal things you already know (or that are wrong). One description of the Waldhausen K-theory of a pointed connected space $X$ is that …

This is the same as the induced map $\pi_4(KO) \to \pi_4(KU)$. Via Bott periodicity (see also this answer), this is the same as the induced map $\pi_0(KSp) \to \pi_0(KU)$ coming from the map sending a …

I'm a bit late to the party, but here's what I suspect the answer should look like. Forgive me for working somewhat to very heuristically throughout. To star things off, here's a silly question: why …

I don't know if this is going to answer your question but here's some relevant background. Splitting idempotents has a very special property from a categorical point of view: it is an absolute colimit …

Complex line bundles over $BG$ are classified by $H^1(BG, \mathbb{C}^{\times}) \cong H^2(BG, \mathbb{Z})$. On the other hand, $1$-dimensional complex representations are classified by $\text{Hom}(G, \ …

Assume that $G$ is connected. Let $T$ be a maximal torus of $G$. Restriction induces a map $R(G) \to R(T)$. Note that $R(T)$ is a Laurent polynomial ring in $r$ variables where $r$ is the rank. Becaus …

The conceptual point is that all of these invariants are Morita invariant because they can be defined directly in terms of the category of modules. Explicitly: Starting from the category of modules …

I feel like we gave up too quickly on parameterized spectra as a reasonable (if necessarily incomplete) answer to this question; we don't get the two formalisms in the question as special cases of an …