Non-additive functors and Euler characteristics (Q5963042)

Let \(\mathcal{A}\) and \(\mathcal{B}\) be abelian categories and assume that \(\mathcal{A}\) has enough projectives. For a (not necessarily additive) functor \(F: \mathcal{A}\to\mathcal{B}\) let \(LF: \mathcal{K}_{\geq 0}(\mathcal{A})\to \mathcal{K}_{\geq 0}(\mathcal{B})\) denote the total derived functor of \textit{A. Dold} and \textit{D. Puppe} [Ann. Inst. Fourier 11, 201--312 (1961; Zbl 0098.36005)], where \(\mathcal{K}_{\geq 0}\) is the homotopy category of chain complexes concentrated in non-negative degrees. Let \(\mathcal{A}_0\) be a weak Serre subcategory of \(\mathcal{A}\) and let \(\mathcal{K}_{\geq 0}^{\mathcal{A}_0}(\mathcal{A})\) denote the full subcategory of \(\mathcal{K}_{\geq 0}(\mathcal{A})\) consisting of those complexes \(X_\bullet\) such that \(H_i(X_\bullet)\in \mathcal{A}_0\) for all \(i\in\mathbb{N}\) and such that there is \(N\in\mathbb{N}\) with \(H_i(X_\bullet)=0\) for \(i\geq N\).NEWLINENEWLINEThe authors show that if \(F: \mathcal{A}\to\mathcal{B}\) as above is of degree \(\leq d\), maps \(\mathcal{K}_{\geq 0}^{\mathcal{A}_0}(\mathcal{A})\) to \(\mathcal{K}_{\geq 0}^{\mathcal{B}_0}(\mathcal{B})\) and satisfies \(F(0)=0\), then there is a unique map \(f: K_0(\mathcal{A}_0)\to K_0(\mathcal{B}_0)\) such that the square NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} \mathcal{K}_{\geq 0}^{\mathcal{A}_0}(\mathcal{A}) \ar[r, "LF"] \ar[d] & \mathcal{K}_{\geq 0}^{\mathcal{B}_0}(\mathcal{B})\ar[d] \\K_0(\mathcal{B}_0) \ar[r, "f" '] & K_0(\mathcal{B}_0) \end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE commutes. Moreover, the function \(f\) is of degree \(\leq d\).NEWLINENEWLINEThis theorem is a generalization of a theorem of \textit{A. Dold} [Math. Ann. 196, 177--197 (1972; Zbl 0221.18007)], which covers the special case where \(\mathcal{A}_0=\mathcal{A}\) and \(\mathcal{A}\) has enough projectives.NEWLINENEWLINEThe authors also give explicit commutations of the map \(f\) for some examples.