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The easy way for me to think about this things is via the Yoneda lemma. This works better with reduced $K$-theory. This is defined for a pointed space $X$ as $$ \tilde K^0 (X) := ker(K^0(X)\to K^0(*)) …

I am not sure if this is going to be a real answer to the question. However I believe these observations might be interesting. Let me briefly sketch a way to describe a $G$-structure in (excessively) …

I claim that for every $A_\infty$-space $A$, there is a canonical $A_\infty$-ring structure on $\Omega^\infty\Sigma^\infty_+A$. First, $\Sigma^\infty_+$ from spaces to spectra is symmetric monoidal. …

Ravenel in his Complex Cobordism and Stable Homotopy Groups of Spheres attributes this result to Stong, in Determination of $H^*(BO(k,⋯,∞),Z_2)$ and $H^∗(BU(k,⋯,∞),Z_2)$, but looking at that paper (wh …

Since the question remains unanswered, let me copy Tom Goodwillie's comment: If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely g …

I do not know of an answer for a general triangulated category (non-topological triangulated categories are very unusual), but as soon as you ask for some more structure the thesis follows very quickl …

You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least …

I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks …

As a first introduction I like these notes by Achim Krause and Thomas Nikolaus. They do require some familiarity with spectra and stable homotopy theory though.