Waldhausen's wS construction of K-theory assigns K-groups to an arbitrary small Waldhausen category, my main goal in reading this construction was to apply it to the case of exact categories with weak equivalencec being isomorphisms. Now my question is, what are the other examples of Waldhausen categories? Are their K-groups serious studied?

Also if I am not mistaken, the category of cofibrant objects in a model category satisfy the axioms of a Waldhausen category, but may not be a small category. Can we talk about the K-groups of some suitable small subcategory of such a category?


The most obvious non-exact examples of Waldhausen categories are just the non-additive ones: a Waldhausen category is only pointed. So Waldhausen's leading example in his original paper is the category $R(X)$ of "retractive" spaces over $X$, that is, retractions $r:Y\to X$ with a choice of a splitting $s:X\to Y$ of $r$. This is pointed, with zero object the identity of $X$, but is certainly not additive, so Waldhausen really needed the extra generality for his goal of defining an algebraic K-theory for topological spaces. This invariant is indeed intensively studied, although to be fair it's possible to define via an additive category given modern techniques of structured ring spectra.

The thing with small categories is not so much to protect against set-theoretical difficulties as to avoid having all K-groups be zero, due to the so-called Eilenberg swindle that becomes possible given countable coproducts. So you do certainly want to restrict to a smaller subcategory. It's most common to use the compact objects, which e.g. gives the Waldhausen category of perfect complexes much used in algebraic geometry.

  • $\begingroup$ Was there a particular problem Waldhausen had in mind for which he wanted to define the K-theory of this category? $\endgroup$ – Arun Kumar Mar 23 '16 at 19:17
  • 2
    $\begingroup$ Waldhausen was interested in calculating the homotopy type of spaces of cobordisms. The space of cobordisms between manifolds over a fixed base $X$ comes as the fiber of the "assembly map" in the Waldhausen $A$-theory, that is, the algebraic K-theory, of $X$. $\endgroup$ – Kevin Arlin Mar 23 '16 at 22:54

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