By Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M

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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. 40 CHAPTER 2. 26 Category (A) . Let A be a category, the default one; in the above discussion we had A = Set .

Dually, an object A is final if, for each object B , there is precisely one morphism from B to A . 10 f: B → A ≡ f = (B) final-Charn Again, mapping ( ) is called the mediator, and it is sometimes written ( → A) to make clear the dependency on A . In typewriter font I would write dem( ), the ‘dual’ of med. By duality, the final object, if it exists, is unique up to a unique isomorphism; the default notation for it is 1 . B . 11 Examples. In Set there is just one initial object, namely the empty set.

The subsets are partly identical (they have some elements in common), but categorically they are different objects. An inclusion A ⊆ A is expressed categorically by the existence of an injective function f : A → A that embeds each element from A into A . ) 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.

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