Weakly enriched categories over a symmetric monoidal category

In this thesis we develop a theory of weakly enriched categories . By 'weakly' we mean an enriched category where the composition is not strictly associative but associative up-to-homotopy. We introduce the notion of Segal enriched categories and of co-Segal categories . The two notions give rise to higher categorical structures. One of the motivations of this work was to provide an alternative notion of higher linear categories, which are known by the experts to be important in both commutative and noncommutative algebraic geometry. The first part of the thesis is about Segal enriched categories. We define such an enriched category as a (colax) morphism of 2-categories satisfying the so called Segal conditions . Our definition is deeply inspired by the notion of up-to-homotopy monoid introduced by Leinster. These weak monoids correspond precisely to Segal enriched categories having a single object. Our work here was to generalize Leinster's work by giving the many object form of his definition. We show that our formalism cover the definition of classical Segal categories and generalizes Leinster's definition. Furthermore we give a definition of Segal DG-category. The theory of enriched categories started with enrichment over a monoidal category. Then the theory was generalized to enrichment over a 2-category, notably by the Australian school. Our formalism generalizes naturally this idea of enrichment over a 2-category by bringing homotopy enrichment at this level. The main results of this work are in the second part of the thesis which is about co-Segal categories. The origin of this notion comes from the fact that Segal enriched categories are not easy to manipulate for homotopy theory purposes. In fact when trying to have a model structure on them, it seems important to require an extra hypothesis that can be too restrictive. We define a co-Segal category as a (lax) morphism of 2-categories satisfying the co-Segal conditions . The idea was to 'reverse' everything of the Segal case i.e from colax to lax, hence the terminology 'co-Segal'. These new structures are much easier to study and to have a homotopy theory of them. The main theorem is the existence of a Quillen model structure on the category of co-Segal precategories; with the property that fibrant objects are co-Segal categories. This model structure is a Bousfield localisation of a preexisting one and lies on techniques which go back to Jardine and Joyal.