Interpretation

Informally, category theory is a general theory of functions. In category theory, morphisms obey conditions specific to category theory itself.

Category theory is founded upon the abstraction of the arrow, f : a → b.

Category theory achieves its wide applicability by describing structure in terms of the existence and properties of arrows.

Categorical descriptions make no assumption about the internal structure of objects; they are given purely in terms of transformation and relations maintained by arrows.

They are data-independent descriptions: the same description may apply to sets, graphs, algebras or anything else that can be considered a category.

A category is a generalisation of a graph with a transitive closure.

CT is very abstract, far more abstract than set theory. Being at a very high level of abstractions, objects in a category are often said to be opaque sets because their internal structure is unknown to us most of the time; perhaps, when working in the category of sets we can know more about the involved sets, but, in general, the internals of objects are opaque. Despite this, we can still infer many facts about a category and its objects just by examining the arrows that relate different objects.

In CT, the properties of objects are inferred from the relations they maintain with other objects, and the relations are realized through the arrows connecting them.

The essence of a category are primarily the arrows that relate one object to another, as well as all the ways the arrows are composed. Arrows reveal relations between objects, they show us how an object is related to other objects in a category and in that way we can gather info about it.