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Tuesday, March 29, 2016

**Abstract:** I will present joint work with Brooke Shipley, in which we have defined a model category structure on the category of $\Sigma^{\infty}X_+$-comodule spectra such that the K-theory of the associated Waldhausen category of homotopically finite objects is naturally weakly equivalent to the usual Waldhausen K-theory of $X$, $A(X)$. I will describe the relation of this comodule approach to $A(X)$ to the more familiar description in terms of $\Sigma^\infty \Omega X_+$-module spectra. I will also explain the construction and properties of the topological coHochschild homology of $X$, which is a potentially interesting approximation to $A(X)$.