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From Sets to Categories (2023)
In this chapter we will see some more set-theoretic constructs, but we will also introduce their category-theoretic counterparts in an effort to gently introduce the concept of a category itself. When we are finished with that, we will try, (and almost succeed) to define categories from scratch, without actually relying on set theory.
Category TheorySet theoryCategory N/A Objects and MorphismsSets and functions N/A ElementBy switching to external diagrams, we lose sight of the particular (the elements of our sets), but we gain the ability to zoom out and see the whole universe where we have been previously trapped in. For example, mathematics deals with a multitude of functions that have the set of numbers as source and target, such as $negate$, $square$, $add one$, and are not at all the identity morphism. Commutativity law is applicable only in contexts where the order is irrelevant i.e. when an object can be represented as the sum of its parts when combined in whichever way.
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