algebraic k theory - Is Higher K-functor the derived functor of K0?

I don't think it's stupid, but I guess it depends what you mean by "derived functor." This is true in the weak sense that K-theory is naturally a space- or spectrum-valued functor, and the $K_i$ is the i-th homotopy of this functor. But it seems not to be the case that $K$-theory is a derived functor in the sense of Cartan-Eilenberg.

Let me discuss the question of the universality of $K$-theory:

I'll abuse terminology and refer to "categories" when I mean categories of a suitable kind, with appropriate added structure --- e.g., exact categories if you want to do Quillen K-theory, Waldhausen categories if you want to do Waldhausen K-theory, Waldhausen $infty$-categories if you want to do K-theory with them, etc. ...

Now if one translates the sense in which $K_0$ is universal as an abelian-group-valued functor on "categories" into the language of stable homotopy theory, one arrives at the universal property satisfied by K-theory as a spectrum-valued functor on "categories."

More precisely, we have additive $K_0$, denoted $K_0^{oplus}$, which is simply the functor that assigns to any "category" $mathcal{C}$ the group completion of the abelian monoid whose elements are isomorphism (or equivalence) classes of objects of $mathcal{C}$, where the sum is $oplus$. This functor is "inadequate" in the sense that there might be some exact (or fiber) sequences of $mathcal{C}$ that $K_0^{oplus}$ cannot see.

To address this, for any "category" $mathcal{C}$, we can build a new "category" $mathcal{E}(mathcal{C})$ whose objects are exact sequences. This "category" admits two functors to $mathcal{C}$ that send an exact sequence $[0to A'to Ato A''to 0]$ to either $A'$ or $A''$. For any functor $F$ from categories to abelian groups, we get an induced homomorphism $Fmathcal{E}(mathcal{C})to Fmathcal{C}oplus Fmathcal{C}$. Let's say that $F$ splits the exact sequences of $mathcal{C}$ if this morphism is an isomorphism, and let's say that $F$ is additive if $F$ splits the exact sequences of every "category."

Now $K_0$ has the following pleasant universal property. It is the initial object in the category of additive functors receiving a natural transformation from $K_0^{oplus}$.

Now to translate all this into stable homotopy. We have additive K-theory, denoted $K^{oplus}$, which is simply the functor that assigns to any "category" $mathcal{C}$ the spectrum corresponding to the group completion of the $E_{infty}$ space given by the (nerve of the) subcategory of $mathcal{C}$ comprised of the isomorphisms (or weak equivalences), where the sum is $oplus$. This functor is again "inadequate" in the sense that there might be some exact (or fiber) sequences of $mathcal{C}$ that $K^{oplus}$ cannot see.

Now for any functor $F$ from categories to spectra, we get an induced homomorphism $Fmathcal{E}(mathcal{C})to Fmathcal{C}vee Fmathcal{C}$. Let's say that $F$ splits the exact sequences of $mathcal{C}$ if this morphism is an equivalence, and let's say that $F$ is additive if $F$ splits the exact sequences of every "category."

Now $K$ has the following homotopy-universal property. It is the homotopy-initial object in the category of additive functors receiving a natural transformation from $K^{oplus}$.

So the universality of K-theory arises not from thinking of the disembodied K-groups, but rather from interpreting K-theory as a spectrum, and rewriting the universal property of $K_0$ in suitably homotopical language.

(References: Gonçalo Tabuada has a paper in which he characterizes K-theory by a similar universal property, and John Rognes and I have begun a similar paper in the context of Waldhausen $infty$-categories, an incomplete draft of which is on my webpage.)

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