A Theorem of the Heart for K-theory of Endomorphisms

Abstract

We show that Quillen's resolution theorem for K-theory also applies to exact $\infty$-categories. We introduce heart structures on a stable $\infty$-category, generalizing weight structures, and using resolution ideas, we show that the category of stable $\infty$-categories equipped with a heart structure fully-faithfully embeds into the category of exact $\infty$-categories. Consequently, we show a generalized theorem of the heart for K-theory, which is equivalent to its invariance under passage to the stable envelope of exact $\infty$-category in the image of the heart functor. Finally, leveraging the above, we show that K-theory of endomorphisms satisfies the theorem of the heart for weight structures, even allowing coefficients in a suitable bimodule.