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from Interlude - A Tour of Math
Published online by Cambridge University Press: 13 October 2022
A first example of a large category of mathematical structures. This means that, instead of looking at an individual structure as a category, we look at all structures of a certain type, and appropriate morphisms between them, and express that as a category. Sets and functions are an essential starting point of mathematics, and one of the fundamental motivating examples of category theory. We start by giving an account of functions that is more aligned with higher level mathematics, and is possibly different from how functions are usually treated in high school. We also examine the total number of possible functions between a given set of inputs and a set of outputs. We then define the identity function, and composition of functions, and check the unit and associativity laws, to show that sets and functions do indeed form a category, which we call Set. Finally, we introduce the idea of sets with extra structure, and the important difference between expressing properties of functions at the level of elements, or at the level of objects and morphisms in the category Set. The latter is the idea of expressing things “categorically”.
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