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
In Appendix A we give formalisations of (most of) the above claims, as well as their formal proofs. With the exception of the part ‘Initiality and colimit as adjointness’, that text uses no other concepts than those known here, so that you may start reading it right now. It is an excellent demonstration of the calculational approach to category theory. 1e Duality Dualisation is a formal manipulation with practical significance. For example, the settheoretic notions of cartesian product and disjoint union are characterised categorically by notions that are each other’s dual.
Convention that in each law the free variables are quantified implicitly in such a way that the well-formedness condition, the premise of init-Type, is met. 38 CHAPTER 2. 22 Application. Here is an example of calculating with initiality: proving that an initial object is unique up to a unique isomorphism. Suppose that both A and B are initial. We claim that ([A → B]) and ([B → A]) establish the isomorphism and are unique in doing so. By init-Self they have the correct typing. We shall show f = ([A → B]) ∧ g = ([B → A]) ≡ f ; g = id A ∧ g ; f = id B , that is, both compositions of ([A → B]) and ([B → A]) are the identity, and conversely the identities can be factored (as in the right-hand side) only in this way.
So, in order to prove that F is left adjoint to G it suffices to establish just one of the statements, and when you know that F is left adjoint to G you may use all of the statements. Before we present the proof of the theorem, we also give some corollaries: additional properties of an adjunction. 4 Theorem. Statements Adjunction, Units, LadAdj, RadAdj, Fusions, and Charns are equivalent. Moreover, the various that are asserted to exist, can all be chosen equal; the same holds for , η, and ε . Adjunction.