Example
Consider the endofunctor sending to . Then the set of natural numbers together with the functions, where and are the obvious functions suggested by their names, is an initial -algebra. The initiality (the universal property for this case) is not hard to establish; the unique homomorphism to an arbitrary F-algebra, for an element of A and a function on A, is the function sending the natural number n to, that is, the n-fold application of f to e.
Read more about this topic: Initial Algebra
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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