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Extra resources for Category Theory [Lecture notes]

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Using the universal mapping property of product twice (see left diagram) we get morphisms λ1 , λ2 : a → a × b such that π1 λ1 = 1a , π1 λ2 = 1a , (3) π2 λ1 = λ1 , π2 λ2 = λ2 : x α γ 1a # α p λ2 y σ1 =ι λ1  1ao /& σ2 =ι

Let (C, F ) be a concrete category. The functor F : C → Set induces a faithful (covariant) functor F : C op → Setop (same object map and same morphism map). 3 can be regarded as a covariant functor P : Setop → Set. Claim: P is faithful. Let α, β : Y → X be morphisms in Setop and assume that α = β. Then the functions α, β : X → Y are not equal, implying that α(x) = β(x) for some x ∈ X. Putting T = Y \{α(x)} ∈ P (Y ) we have x ∈ / α−1 [T ] = P (α)(T ), but x ∈ β −1 [T ] = P (β)(T ). Hence P (α)(T ) = P (β)(T ), yielding P (α) = P (β).

Indeed, from (3) we have π1 λi α = 1a α = α, π 2 λi α = λi α = β (i = 1, 2). So by uniqueness, λ1 α = λ2 α. Therefore, (x, (α, α)) ∈ Dpb . This yields a unique morphism γ : x → p such that σi γ = α (i = 1, 2) or, equivalently, such that ιγ = α. The claim follows and the proof is complete. ” Let C be a category and let λ1 : a → b1 and λ2 : a → b2 be two morphisms in C: λ2 / b2 . a λ1  b1 Form an auxiliary category D = Dpo as follows: Take for objects pairs (x, (α1 , α2 )), where x is an object of C and αi : bi → x (i = 1, 2) are morphisms in C such that α1 λ1 = α2 λ2 , that is, such that the following diagram is commutative: a λ1 λ2 α2  b1 / b2 α1 37 / x ; take as morphisms from the object (x, (α1 , α2 )) to the object (y, (β1 , β2 )) all morphisms γ : x → y in C such that γαi = βi (i = 1, 2), that is, such that the following diagram is commutative: a λ1 λ2 α2  b1 / b2 α1 / x β2 γ  / y; β1 and define composition of morphisms to be the composition in C.

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