"Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows." (Saunders Mac Lane, "Categories for the Working Mathematician", 1971)
"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)
"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)
"Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set – called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation. No doubt, this is the simplest definition of a model that, unfortunately, plays a limited role in scientific applications of mathematics. Because applications exhibit a need for a large variety of vastly different mathematical structures – some topological or smooth, some algebraic, order-theoretic or combinatorial, some measure-theoretic or analytic, and so forth, no useful overarching definition of a mathematical model is known even in the edifice of modern category theory. It is difficult to come up with a workable concept of a mathematical model that is adequate in most fields of applied mathematics and anticipates future extensions."
"Category theory studies relationships between things and builds on this in various ways: characterising things by what properties they have, finding the pond in which things are the biggest fish, putting things in context, expressing subtle notions of things being ‘more or less the same’." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"In category theory there is always a tension between the idealism and the logistics. There are many structures that naturally want to have infinite dimensions, but that is too impractical, so we try and think about them in the context of just a finite number of dimensions and struggle with the consequences of making these logistics workable."
"This is how category theory arose, as a new piece of maths to study maths. In a way category theory is an ultimate abstraction. To study the world abstractly you use science; to study science abstractly you use maths; to study maths abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
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