A Gentle Introduction to Category Theory - the calculational by Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M

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

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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.

22 below); the default notation for it is 0 . An alternative notation for ([0 → B]) is ¡B . 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 .

So, and are natural transformations 29 1E. DUALITY “of a higher type”, and the omission of the subscripts to our convention for natural transformations. and thus falls under 3. If F is left adjoint to G , or, equivalently, G is right adjoint to F , then F preserves colimits (such as initial objects and sums; all these notions will be defined later), and G preserves limits (such as final objects and products, again to be defined later). 51 More on adjointness. In Appendix A we give formalisations of (most of) the above claims, as well as their formal proofs.

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