The \(K\)-theory of endomorphisms of spaces
The author gives an \(\mathrm{End}\)-\(\mathrm{Nil}\) description of Waldhausen algebraic \(K\)-theory of spaces. The set-up is as follows: Let \(M_\bullet\) be a monoid and denote by \(M\) its realization. Let \(\mathbb{T}(M)\) be the category of based spaces with a left action by \(M\) and \(\mathbb{C}(M)\) be the subcategory of \(\mathbb{T}(M)\) consisting of cofibrants. There are several categories related to these: the category of \textit{finite} objects in \(\mathbb{C}(M)\), that of \textit{homotopy finite} objects, and \textit{finitely dominated} objects, these are related as follows: NEWLINE\[NEWLINE\mathbb{C}_f(M)\subset \mathbb{C}_{hf}(M)\subset \mathbb{C}_{fd}(M)\subset \mathbb{C}(M)\subset \mathbb{T}(M).NEWLINE\]NEWLINE The author gives the definition of the endomorphism category, \(\mathrm{End}_?^S(M)=\mathrm{End}_?^S(*,M)\) ``localized'' with respect a to a multiplicative set \(S\). The author proves that \(\mathrm{End}_?^S(*,M)\) is a Waldhausen category and the main theorem is {Theorem.} Let \(?=f, hf, fd\) denote one of the above defined categories, then there is a homotopy equivalence of spectra NEWLINE\[NEWLINE\widetilde{K}(\mathrm{End}^S_?(*,M))\simeq \Omega E^S_?A(*,M),NEWLINE\]NEWLINE where \(A^?(*,M)=\Omega|wS_\bullet\mathbb{C}_?(M)|\), and the latter is Waldhausen \(S\)-construction to the respective subcategories of weak equivalences.
scientific article; zbMATH DE number 6618802