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, 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, Bull. of the Belgian Math. Soc. Simon Stevin 20 (2013), 1-8.

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, in Topological Complexity and Related Topics, Contemporary Mathematics, vol. 702, Amer. Math. Soc., Providence, RI, 2018, pp. 91-101.