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
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. 55 Corollary. For each definition expressed in the categorical language, of a concept or construction xxx, you obtain another concept, often called co-xxx if no better name suggests itself, by dualising each term in the definition.
A : A →A GF A εB : ψ: F A →B B ψ A,B : A →A GB ∈ : : Objects of B B →B B F A →B B (“to B ”) depend on F, G and have the F GB →B B ϕ: A →A GB ϕ A,B : F A →B B Mappings and are called lad and rad, respectively, from left adjungate and right adjungate. As a memory aid: the first symbol of has the shape of an ‘L’ and therefore denotes lad. In typewriter font I would write lad( ) and rad( ). For readability I will omit the typing information whenever appropriate, as well as most subscripts. Omitting the subscripts is dangerous (and even erroneous) if categories A and B are built upon another one.
55 it is shown that left adjoints preserve colimits. 50 CHAPTER 2. 40 Initiality as colimit. Let A be a category, the default one. Take D empty, so that D: D → A is the empty functor. Then a cocone δ for D is the empty family ( )B of morphisms, where B = tgt δ . Suppose γ = ( )A is a colimit for D ; it may or may not exist. Then A is initial in A . To show this, we establish init-Charn, constructing ([ ]) along the way. For arbitrary object B and morphism x , x: A → B ≡ property of the empty natural transformations ( )A and ( )B ( )A ; x = ( ) B ≡ γ = ( )A is colimit for D , and ( )B is cocone for D ; \ -Charn x = γ\( )B ≡ define ([B]) = γ\( )B x = ([B]).