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Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this spectra. I guess that stable homotopy groups are known to be representable this way? Can you help me to get an overview?

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Thomason's paper "Symmetric monoidal categories model all connective spectra" claims to show exactly what the title claims - namely, you can model all generalized homology theories E with En(*) = 0 for n < 0 by taking the spectrum associated to a symmetric monoidal category. So in principle, these are what you might feel like you should get.

However, Waldhausen categories are more restrictive - they require that the symmetric monoidal structure is actually the underlying categorical coproduct. I don't know of any results along this line.

It appears that Thomason's proof is something like the category of weakly contractible spaces over X, but - I will be blunt - I have never managed to sort through Thomason's paper. It seems conceivable that the object he constructs might be equivalent to something coming from a Waldhausen category or its opposite, but this might be optimistic.

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Thanks for the nice reference. I think there might be a good chance to build a Waldhausen category that models Thomason's construction. Does anyone know how to fill the gap? – user2146 Dec 19 '09 at 12:53

The Waldhausen category C which is the category of finite pointed sets, with cofibrations the monomorphisms and weak equivalences the isomorphisms, has K(C) = the sphere spectrum—or so I am told.

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Oblig: This is called the Barrat-Priddy-Quillen theorem. – Tyler Lawson Dec 19 '09 at 3:54

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