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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We shall briefly define the notion of colimit, and present its calculational properties derived from the characterisation by initiality. We shall also give a nontrivial application involving colimits. By definition, limits are dual to colimits. So by duality limits generalise such notions as product, equaliser, and several others. Each limit is a certain final object, and each final object is a certain limit. The formal definition uses the notions of a diagram D and of the cocone category D , which we now present.

Here is one; don’t try to understand what it means, we’ll meet it in the sequel. Apart from dualising the statement, 30 CHAPTER 1. THE MAIN CONCEPTS we also rename some bound variables and interchange the sides of the left-hand equation (which doesn’t affect the meaning). ∃([ ]) ∀B ∀f : A → B ∀ϕ: F B → B :: α ; f = F f ; ϕ ≡ f = ([ϕ]) ∃ ( ) ∀B ∀g: B → A ∀ψ: B → F B :: ψ ; F g = g ; α ≡ g = (ψ) . Exercise: infer the typing of F, α, ([ ]), and ( ) in these formulas. Notice that the type of the free variable α changes due to the dualisation.

Exercise: spell out in terms of B what it means for two functors from A to B to be isomorphic as objects in Ftr (A, B) . ) Are functors II Seq and Seq II isomorphic in Ftr (Set , Set ) ? Exercise: prove that the composition of isomorphisms is an isomorphism again. What is the inverse of a composite isomorphism? Exercise: prove that each isomorphism is both monic and epic. Exercise: given that A is a subcategory of B , prove that each monomorphism in B is monic in A . Exercise: prove in Set that a function is epic iff it has a pre-inverse.

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