Grothendieck définit d'abord les univers, puis les types de diagrammes avant de définir une catégorie
By providing a system in which all the usual mathematical concepts can be expressed rigorously, Set Theory has represented the first serious attempt of Logic to unify Mathematics at least at the level of language.
Later, Category Theory offered an alternative abstract language in which most of Mathematics can be formulated and, as such, has represented a further advancement towards the goal of 'unifying Mathematics'.
Anyway, both these systems realize a unification which is still limited in scope, in the sense that, even though each of them provides a way of expressing and organizing Mathematics in one single language, they do not offer by themselves effective methods for an actual transfer of knowledge between distinct fields.
The kind of unification realized by these theories can be considered static, in the sense that it is achieved through a process of generalization, which allows to regard different concepts as particular cases of a more general one but does not offer by itself a way for transferring information between them:
For example, the fact that both preorders and groups are particular instances of the general notion of category does not give by itself a means for transferring results about preorders to results about groups or conversely.
If a category object form a set, then the category is called small.
If they don't form a set, then it's a large category.
Morphisms between two categories form a set.
In higher-order categorie, arrows don't form sets, they form objects in a category.
Category Theory 1.2: What is a category? (28:31)
- Quel est le lien entre les n-catégories et les topos d'ordre supérieur ?
- Ce dessin représente-t-il des n-catégories ?