LS category: Product formulas
Let \(f:X\to Y\) be a continuous map between simply connected CW complexes. Denote by \(q_n:G_n(Y)\to Y\) the \(n\)th Ganea fibration of \(Y\) and \(\varphi:(T(V),d) @>\simeq>> C^*(Y;k)\) a semifree model of the cochain algebra on \(Y\). The category of \(f\) is the least integer \(n\) such that \(f\) factors up to homotopy through \(q_n\), and, for a field \(k\), the module category \(M\text{cat}_k (f)\) is the least integer \(n\) such that \(C^*(f,k) \circ\varphi\) factors up to homotopy through \(C^*(q_n)\) as a \((T(V),d)\)-module. Extending results of \textit{Y. Félix}, \textit{S. Halperin} and \textit{J.-M. Lemaire} [Topology 37, No. 4, 749-756 (1998; Zbl 0897.55001)], the author proves that \(M \text{cat}_k(f\times g)= M \text{cat}_k(f) +M \text{cat}_k(g)\). This result extends the original result because \(M \text{cat}_Q(\text{id}_X)=\text{cat}(X_0)\) (K. Hess).
scientific article; zbMATH DE number 1501768