Algebraic K-theory is an invariant defined on categories that records how object in the category are related by exact sequences --- it is a homotopical version of the classical Euler characteristic. However, there are many categories of interest that do not have exact sequences, but instead have cutting and pasting operations. For example, the category of varieties or the category of polytopes. I'll describe how to define a higher algebraic K-theory for categories like this, and show that it's not so different from the case of more algebraic categories. Even better, theorems like Quillen's Devissage and Localization can be proved internal to these structures. Time permitting, I'll describe how the cutting and pasting of polytopes is intimately related to the weight filtration on the algebraic K-theory of fields.

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Location: ESB 4133 (PIMS Lounge) Jonathan Campbell, Vanderbilt