L.S.-category in a model category (Q1208230)

The notion of category was introduced by \textit{L. Lusternik} and \textit{I. Schnirelmann} [Méthodes topologiques dans les problèmes variationnels (1934; Zbl 0011.02803)] to study the stationary points of functions on a manifold. For a path-connected, normal and categorically well based space, \(X\), the LS-category can be defined by two equivalent ways: (1) as the least integer, \(n\), such that there exists a (homotopy) factorization of the diagonal, \(\Delta: X\to X^{n+1}\), through the fat wedge \(T^ n(X)\) of \(X\), [\textit{G. W. Whitehead}, Centre Belge Rech. math., Colloque de Topologie algébrique, Louvain les 11, 12 et 13 juin 1956, 89-95 (1957; Zbl 0079.391)], (2) or as the least integer, \(n\), such that Ganea's fibration, \(G_ n X\to X\), admits a section up to homotopy [\textit{T. Ganea}, Illinois J. Math. 11, 417-427 (1967; Zbl 0149.407)]. Fat wedge and Ganea's fibrations can be realized in Quillen's closed model category but, in general, the previous definitions give two distinct integers as that can be noted for the dual notion of cocategory [Ganea, loc. cit.]. In this paper the author introduced the notion of \(J\)-category, as a closed model category enriched with a cube axiom (which is not autodual!). The author proves that the Whitehead and Ganea definitions coincide in a \(J\)-category, \(C\), and calls \(C\)-cat this common notion. He studies the basic properties of \(C\)-cat, proves an intrinsic version of the mapping theorem of \textit{Y. Félix} and \textit{S. Halperin} [Trans. Am. Math. Soc. 273, 1-37 (1982; Zbl 0508.55004)] and gives examples of \(J\)-categories. Finally, if \(F: C\to D\) is an ``appropriate'' functor between \(J\)-categories, the author shows that \(C\)-\(\text{cat}(X)\), \(X\in C\), can be approximated by \(D\)- \(\text{cat}(F(X))\).