By Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M
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Extra info for A Gentle Introduction to Category Theory - the calculational approach
56 Examples. We shall meet many examples in the sequel, notably the examples mentioned in the introduction to this section. 24) is obtained by dualising each notion of A , that is, an object in Aop a morphism in Aop f : A → B in Aop f ;Aop g id Aop ,A is: is: ≡ = = dual A for some object A in A dual f for some morphism f in A dual(f : A → B) in A dual(f ;A g) dual(id A,A ) . It follows that (Aop )op = A , and that the dual of a statement holds for A if and only if the statement itself holds for Aop .
These are referred to by the hint ‘product’ or ‘sum’. Here is a list; some of these are just the laws presented before. f × g ; exl f ∆ g ; exl f × g ; exr f ∆ g ; exr f ;g∆h exl ∆ exr (h ; exl ) ∆ (h ; exr ) f ∆g ;h×j f ×g ;h×j f ∆g =h∆j = = = = = = = = = ≡ exl ; f f exr ; g g (f ; g) ∆ (f ; h) id h (f ; h) ∆ (g ; j) (f ; h) × (g ; j) f =h∧g =j inl ; f + g inl ; f ∇ g inr ; f + g inr ; f ∇ g f ∇g ;h inl ∇ inr (inl ; h) ∇ (inr ; h) f +g ;h∇j f +g ;h+j f ∇g =h∇j = = = = = = = = = ≡ f ; inl f g ; inr g (f ; h) ∇ (g ; h) id h (f ; h) ∇ (g ; j) (f ; h) + (g ; j) f =h∧g =j Exercise: identify the laws that we’ve seen already, and prove the others.
Sometimes the proof has the following format: ⇒ f: A → B 37 2B. INITIALITY AND FINALITY ⇒ .. f = expression not involving f ⇒ . ⇒ f : A → B. = ([B]) , by suitably defining ([ ]) In this case we say that we establish the equivalence init-Charn by circular implication. In general the formulas are not as simple as suggested in the above calculations, since mostly we will be dealing with initiality in categories built upon another one, so that the typing f : A → B is a collection of equations in the underlying category.
A Gentle Introduction to Category Theory - the calculational approach by Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M