Relation between $K$-theories

Question

I apologize in advance if this question is too vague/general.

I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, operator $K$-theory, and possibly some others I haven't heard of).

My first guess was that they would have turned out to be equivalent in a similar manner that ordinary cohomology theories are equivalent (for suitably nice spaces), but this obviously can't be the case because the $K$-functors of the different $K$-theories aren't even defined on the same category. So then, what is the relation between the different $K$-theories. Is there any distinguishing property all these theories possess which makes them all deserving of the same title?

Answer 1

Swan's theorem tells you that the category of finite dimensional topological vector bundles on a compact topological space $X$ is equivalent to the category of finite type projective $A$-modules with $A = C^0(X)$ simply by considering global sections. This is the key point explaining why topological $K_0$ relates at least formally to $K_0$ of operator algebras and algebraic $K_0$ of rings.

Answer 2

Johnny, you might be looking for some kind of "universal property" that $K$-theory satisfies. This is recent work of Blumberg, Gepner and Tabuada where they proved that (connective) algebraic $K$-theory is, in some sense, the universal additive (satisfying a categorified version of additivity of Euler charactetristic) invariant of inputs like Waldhausen categories (roughly, a category with a notion of quotients) - the paper is: http://arxiv.org/pdf/1001.2282.pdf

Answer 3

One could ask the same question about e.g. cohomology - there's all sorts of things that get called cohomology that aren't obviously equivalent to each other in any reasonable sense. I think the historical answer is that people see familial resemblances and name things based on those. In general it seems odd to me to expect more coherence from names than this. People name things for all sorts of reasons.

Anyway, here is my very sketchy understanding of the history: things in some sense started with Grothendieck, who defined algebraic $K_0$ in terms of coherent sheaves for the sake of Grothendieck-Riemann-Roch. On a smooth variety one can also use vector bundles. Atiyah and Hirzebruch noticed that the definition involving vector bundles could be imitated for spaces and so defined topological $K_0$ by analogy, and furthermore noticed that Bott periodicity implied that topological $K_0$ could be extended to an extraordinary cohomology theory; this was for the sake of the index theorem.

With this result in mind it's tempting to go back and try to extend algebraic $K_0$ to a cohomology theory in some sense and this was finally done by Quillen. In a different direction, Swan's theorem implies that for compact Hausdorff spaces $K_0$ can be defined in terms of finitely generated projective modules over the C*-algebra of continuous functions, and this suggests an extension to noncommutative C*-algebras which gave rise to operator K-theory (I don't know who the relevant authors in this case are though).

So historically I think the connection is clear:

K-theory is something you build out of vector bundles, suitably interpreted.

But nowadays lots of things are called K-theory that don't look anything like this, so here is a more modern perspective. Roughly speaking,

K-theory is a fancier version of the group completion of a monoid.

You can see this already in how ordinary K-theory is defined in terms of adjoining inverses to the operation of direct sum on (isomorphism classes of) vector bundles. There are (at least) two fancy things to do here and I only understand one of them (and not all that well) so let me explain that one.

Let $C$ be a symmetric monoidal groupoid. The underlying groupoid $C$ can be turned into a space called the geometric realization of its nerve $N(C)$. (For a discrete groupoid this is just a disjoint union of classifying spaces of groups but in general it's interesting to run this construction for groupoids enriched in topological spaces, say.) The symmetric monoidal structure transports to a structure of some sort on this space. One would like to say it turns the space into a topological commutative monoid but this statement is too strong. The correct statement is that it turns the space into a "homotopy commutative monoid," or more precisely an $E_{\infty}$-monoid. One can adjoin inverses to this monoid in a suitably homotopical way and the result is a spectrum, the K-theory spectrum of $C$.

The main thing you need to know here about spectra is that on the one hand they are homotopical versions of abelian groups and on the other hand they represent homology and cohomology theories, such as K-theory! Two examples:

• The free symmetric monoidal groupoid on an object is the groupoid of finite sets equipped with disjoint union. The K-theory spectrum of this guy is the free spectrum on a point, or the sphere spectrum. This is the spectrum that represents the homology theory given by taking the stable homotopy groups of a space. (I've stated this fact in a way that makes it look like it should follow straightforwardly from universal properties but it doesn't; this is the Barratt-Priddy-(Quillen-(Segal?)) theorem.)
• Let $k = \mathbb{R}, \mathbb{C}$ and consider the groupoid of finite-dimensional $k$-vector spaces equipped with direct sum, with the morphisms topologized in the usual way. The K-theory spectrum of this guy is the spectrum representing real resp. complex (connective?) K-theory. If we didn't topologize the morphisms we would get algebraic K-theory instead.