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from Part One - Building up to Categories
Published online by Cambridge University Press: 13 October 2022
A first look at relations more abstractly. We begin by thinking about more relaxed notions of equality and observe that, if we relax the notion too much, then we might get some undesirable consequences. This motivates the concepts of reflexivity, symmetry, and transitivity. We’ll explore these properties for various relations, including informal ones such as “is at school with” and “is a friend of”, and then more formal mathematical ones such as “divides” and “is congruent modulo n”. We introduce equivalence relations, which are those satisfying all three properties, and observe that many relations are not equivalence relations but are nonetheless still interesting and worthy of study. This is to motivate the more relaxed axioms for a category. This chapter starts using more formal notation.
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