By Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M
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Additional info for A Gentle Introduction to Category Theory - the calculational approach
17 Proving initiality. One may prove that an object A is initial in the category, by providing a definition for ([ ]) and establishing init-Charn. Almost every such a proof in the sequel has the following format. For arbitrary f and B , f: A → B ≡ . ≡ f = an expression not involving f ≡ define ([B]) = the right-hand side of the previous equation f = ([B]). Actually, the last two lines in the calculation are superfluous: the remaining lines clearly show that the statement f : A → B has precisely one solution for f .
Prove in each case the above claims. Would you call A ∪ B a disjoint union of A and B ? Why, or why not? 24 inl ; h = f ∧ inr ; h = g . This is an entirely categorical formulation. In addition, the formulation suggests to look for a characterisation by means of initiality (or finality). 25 h: (inl , inr ) → (A,B) (f, g) . So (inl , inr ) is initial in (A, B) . Having available the pair (inl , inr) (as ‘the’ initial object in (A, B) ), the set A + B can be defined by A + B = tgt inl = tgt inr . Thus the notion of disjoint union has been characterised categorically, by initiality, and it turns out that the injections inl , inr and operation ∇ are as relevant for the notion of disjoint union of A and B as the set A + B itself.
So the sequence of embeddings is expressed by the diagram: A0 • f0 ✲ A1 • f1 ✲ A2 • ··· Each composition fi ; fi+1 ; · · · ; fj−1 : Ai → Aj denotes the accumulated embedding of Ai into Aj . Now consider a cocone δ for that diagram: 52 CHAPTER 2. CONSTRUCTIONS IN CATEGORIES A0 • f0 ✲ A1 • f1 ✲ A2 • ··· B • ✻ δ δ0 ✫ δ1 ✫ δ2 ✫ ✪ The equalities fi ; δi+1 = δi imply that fi ; fi+1 ; · · · ; fj−1 ; δj = δi . So each element from each of the subsets is mapped “unambiguously” into B via δ . ) Let A = (n :: An ) , the infinite union of all the subsets, and let γi be the embedding of Ai in A in such a way that each element of each of the subsets is mapped “unambiguously” (as explained for δ above) into A .
A Gentle Introduction to Category Theory - the calculational approach by Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M