"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)
"Category theory has developed classically, beginning with definitions and axioms and proceeding to a long list of theorems. Category theory is not topology (and so will not be described here), but it can be used to understand some of the relationships that exist among classes of topological spaces. It can be used to bring unity to diversity. [...] the theory of categories is not complete, it may not be completable, but it is a step forward in understanding foundational questions in mathematics." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)
"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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