If a category C satisfies an appropriate set of axioms, closely related to those which hold in a Quillen model category, we show that it is possible to construct objects in C which are analogous to certain constructions of Ganea and Whitehead for topological spaces. Given these generalized constructions in C, we are able to define a notion of Lusternik-Schnirelmann category, internal to C, C-cat. We state properties of C-cat and examine the relationship between C-cat and D-cat, whenever there is an appropriate functor C->D
J.-P. Doeraene, L.S.-category in a model category, Journal of Pure and Applied Algebra 84 (1993) 215-261.
Dans LS category in a model category, il est montré que les présentations de Whitehead et Ganea de la LS-catégorie coïncident dans une catégorie à modèles fermée qui est propre et satisfait l'axiome du cube. Nous nous affranchissons ici de ces deux restristions, d'abord en définissant les produits fibrés homotopiques sans propreté, et ensuite en remplaçant l'axiome du cube par une caractérisation au niveau des morphismes.
J.-P. Doeraene, D. Tanré, Axiome du cube et foncteurs de Quillen, Ann. Inst. Fourier, 45-4 (1993), 1061-1077.
We show here how the techniques based on homotopy pull backs and push outs lead to simple proofs for apparently difficult (known or unknown) results. They can be used not only in the category of topological spaces, but also in any Quillen's model category. Many of them rely on the two `join theorems' we prove here. Further applications are the study of holonomy, or of the Lusternik-Schnirelmann category.
J.-P. Doeraene, Homotopy pull backs, homotopy push outs and joins, Bull. of the Belg. Math. Soc. Simon Stevin 5-1 (1998), 15-37.
Most of the properties of the category of Lusternik-Schnirelmann come from the cube theorems of Mather, especially the second. The duals of these theorems are false, which makes the dual category problematic. However, a weaker version of the second cube theorem is sufficient to get the usual properties of the LS-category. In this note, we show that this weaker version of the dual of the first theorem is false. The weaker version of the dual of the second cube theorem remains an open problem.
J.-P. Doeraene, M. El Haouari, About dual cube theorems, Topology and its Applications 142/1-3 (2004), 61-72.
The Lusternik-Schnirelmann category has been described in different ways. Two major ones, the first of Ganea, the second of Whitehead, are presented here with a various number of variants. The equivalence of these variants rely on the axioms of a Quillen's model category, but also sometimes to an additional axiom, namely the so-called `cube axiom'.
J.-P. Doeraene, M. El Haouari, The Ganea and Whitehead variants of the Lusternik-Schnirelmann Category, Canad. Math. Bull. 49-1 (2006), 41-54.
This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category S-Top of stratified spaces, that are topological spaces X endowed with a partition F and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,F) of S-Top together with a class A of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe. If (X,F) in S-Top, we define a transverse subset as a subspace A of X such that the intersection S inter A is at most countable for any S in F. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a C1-foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.
J.-P. Doeraene, E. Macias-Virgós, and Daniel Tanré, Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations, Topology and its Appl. 157 (2010), no. 9, 1680-1689.
James' sectional category and Farber's topological complexity are studied in a general and unified framework. We introduce `relative' and `strong relative' forms of the category for a map. We show that both can differ from sectional category just by 1. A map has sectional or relative category less than or equal to n if, and only if, it is `dominated' (in a different sense) by a map with strong relative category less than or equal to n. A homotopy pushout can increase sectional category but neither homotopy pushouts, nor homotopy pullbacks, can increase (strong) relative category. This makes (strong) relative category a comfortable tool to study sectional category. We completely determine the sectional and relative categories of the fibres of the Ganea fibrations. As a particular case, the `topological complexity' of a space is the sectional category of the diagonal map. So it can differ from the (strong) relative category of the diagonal just by 1. We call the strong relative category of the diagonal `strong complexity'. We show that the strong complexity of a suspension is at most 2.
J.-P. Doeraene, M. El Haouari, Up to one approximations of sectional category and topological complexity, Topology and its Appl. 160 (2013), 766–783.
In the previous paper the authors introduced a relative category for a map that differs from the sectional category by just one. The relative category has specific properties (for instance a homotopy pushout does not increase it) which make it a convenient tool to study the sectional category. The question to know when secat equals relcat arises. We give here some sufficient conditions. Applications are given to the topological complexity, which is nothing but the sectional category of the diagonal.
J.-P. Doeraene, M. El Haouari, When does secat equal relcat ?, Bull. of the Belgian Math. Soc. Simon Stevin 20 (2013), 1-8.
The classical Hopf invariant is defined for a map f: Sr -> X. Here we define `hcat' which is some kind of Hopf invariant built with a construction in Ganea's style, valid for maps not only on spheres but more generally on a `relative suspension' f: ΣA W -> X. We study the relation between this invariant and the sectional category and the relative category of a map. In particular, for f being the `restriction' of f on A, we have relcat(i) ⩽ hcat(f) ⩽ relcat(i) + 1 and relcat(f) ⩽ hcat(f).
J.-P. Doeraene, M. El Haouari, Yet another Hopf Invariant, arXiv:1501.03712
We first compute James’ sectional category (secat) of the Ganea map 𝑔𝑘 of any map 𝜄𝑋 in terms of the sectional category of 𝜄𝑋: we show that secat 𝑔𝑘 is the integer part of secat 𝜄𝑋 ⁄ (𝑘 + 1). Next we compute the relative category (relcat) of 𝑔𝑘. In order to do this, we introduce the relative category of order 𝑘 (relcat𝑘) of a map and show that relcat 𝑔𝑘 is the integer part of relcat𝑘 𝜄𝑋 ⁄ (𝑘 + 1). Then we establish some inequalities linking secat and relcat of any order: we show that secat 𝜄𝑋 ⩽ relcat𝑘 𝜄𝑋 ⩽ secat 𝜄𝑋 +𝑘+1 and relcat𝑘 𝜄𝑋 ⩽ relcat𝑘+1 𝜄𝑋 ⩽ relcat𝑘 𝜄𝑋 + 1. We give examples that show that these inequalities may be strict.
J.-P. Doeraene, Sectional Category of the Ganea Fibrations and Higher Relative Category, Chinese Journal of Mathematics, Volume 2016, Article ID 8320742, 5 pages
We propose a definition of `sectional category of a class of maps’. This combines the notions of `sectional category’ of James, and `category of a class of spaces’ of Clapp and Puppe.
Jean-Paul Doeraene, Mohammed El Haouari, Carlos Ribeiro, Sectional category of a class of maps, Topological Complexity and Related Topics, Contemporary Mathematics, vol. 702, Amer. Math. Soc., Providence, RI, 2018, pp. 91-101.