"It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation […]. The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether […]" (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"The theory [of categories] also emphasizes that, whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these new objects as an integral part of their definition. The pursuit of this program entails a simultaneous consideration of objects and their mappings (in our terminology, this means the consideration not of individual objects but of categories). This emphasis on the specification of the type of mappings employed gives more insight onto the degree of invariance of the various concepts involved." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"The notion of an abstract group arises by consideration of the formal properties of one-to-one transformations of a set onto itself. Similarly, the notion of a category is obtained from the formal properties of the class of all transformations y : X → Y of any one set into another, or of continuous transformations of one topological space into another, or of homomorphisms, of one group into another, and so on." (Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56, 1950)
"Category theory is an embodiment of Klein’s dictum that it is the maps that count in mathematics. If the dictum is true, then it is the functors between categories that are important, not the categories. And such is the case. Indeed, the notion of category is best excused as that which is necessary in order to have the notion of functor. But the progression does not stop here. There are maps between functors, and they are called natural transformations." (Peter Freyd, "The theories of functors and models", 1965)
"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)
"The theory of categories has arisen in the last twenty-five years and now constitutes an autonomous branch of mathematics. It owes its origin and early inspiration to developments in algebraic topology. When the basic concepts of category, functor, natural transformation and natural equivalence were first formulated by Eilenberg and Mac Lane they served immediately to provide the appropriate framework for describing the way in which algebraic tools were used, and could be used, in the study of topology. It was surely evident from the outset, to the inventors of these fundamental notions and to others, that their domain of application certainly extended far beyond that of algebraic topology. […] However, it was not clear in the early stages that there was a “pure” theory latent within the domain of categories and functors which was capable of assuming substantial proportions within the body of mathematics. […] Nevertheless, it is only in the last ten years, or less, that the source of inspiration for advances in category theory has come to any considerable extent from within the theory itself." (Ion Bucur & Aristide Deleanu, "Introduction to the Theory of Categories and Functors", 1968)
"Yet, unless I am not sufficiently aware of current trends, the set-theoretical difficulties in handling categories have not inspired many set theorists and it has had little impact in logic as a whole. Thus, although we thoroughly accepted highly impredicative set theory because we understand its internal cogency, we, as logicians, are less likely to accept category theory whose roots lie in algebraic topology and algebraic geometry. It could be retorted that the existing axioms of infinity are ample to cover formalizations of category theory, yet an obstinate categorist could say that categories themselves should be accepted as primitive objects." (Paul J Cohen, "Comments on the foundations of set theory", 1971)
No comments:
Post a Comment