Structures algébriques n-aires généralisant les catégories. (n- ary algebraic structures generalizing categories) (Q1262937)

In the first chapter the author gives a general algebraic definition of (normal) categories as special partial algebras and four different generalizations of n-categories, where in the case of \(n=2\) these become normal categories. These definitions are ``double categories'', which are generalized to E-n-categories, V-categories, T-n-categories and H-n- categories. In the second chapter the author gives some general results on the considered categories and the forgetful functor \(C\to SET\). The third chapter contains embeddings of the categories in some categories of mappings. Can any n-category in some sense be imbedded in a category derived on a 2-category - like any n-group can be imbedded in a n-group derived by a 2-group (cf. Post ``coset theorem'', 1940)? This question is treated in the fourth chapter. In the fifth chapter it is shown that any n-category can be represented by a category of n endomorphisms such that n-morphisms are products of n morphisms of the category. Finally four open problems in the theory of n-categories are posed. The paper contains a list of 107 papers on different kinds of n- categories.