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 category theory this property is called a ‘middle exchange rule’ or ‘interchange rule’. Here is a proof that ∇ and ∆ abide: (f ≡ ≡ ≡ ≡ g) ∆ (h ∇ j) = (f ∆ h) ∇ (g ∆ j) ∇ -Charn [f, g, h := lhs, f ∆ h, g ∆ j] inl ; (f ∇ g) ∆ (h ∇ j) = f ∆ h ∧ inr ; (f ∇ g) ∆ (h ∇ j) = g ∆ j ∆ -Fusion (at two places) (inl ; f ∇ g) ∆ (inl ; h ∇ j) = f ∆ h ∧ (inr ; f ∇ g) ∆ (inr ; h ∇ j) = g ∆ j ∇ -Self (at four places) f ∆h=f ∆h ∧ g∆j =g∆j equality true. ∇ Exercise: give another proof in which you start with ∆ -Charn rather than ∇ -Charn.
It follows that x: tgt γ → tgt δ . The condition on x can be written simply γ ; x = δ , when we define the composition of a cocone with a morphism by: (γ ; x)A = γA ; x . (composition of a cocone with a morphism) Then γ: D → . C and x: C → C ⇒ γ ; x: D → . C . (An alternative would be to write γ ; x , since for x: C → C the constant function x = A → x is a natural transformation of type C → . 38 Definition. Let A be a category, the default one, and let D diagram in A . A colimit for D is: an initial object in D ; it may or may not exist.
33 Additional laws. The following law confirms the choice of notation once more. \-Compose p\q ; q\r = p\r Here is one way to prove it. p\q ; q\r = \-Fusion p\(q ; q\r) = \-Self p\r. , p\f,g q and q\h,j r , and q is not necessarily a coequaliser of f, g . 34 ([A → B])A ; ([B → C])B = ([A → C])A init-Compose where A and B are full subcategories of some category C and objects B, C are in both A and B ; in our case A = (f g) , B = (h j) , and C = (D) where D is the common target of f, g, h, j . Then the proof runs as follows.
A Gentle Introduction to Category Theory - the calculational approach by Fokkinga, M.M.; Jeuring, J.T.; Fokkinga, Maarten M