By Yun Fan

This quantity is predicated at the lectures given by means of the authors at Wuhan collage and Hubei college in classes on summary algebra. It provides the elemental strategies and easy homes of teams, jewelry, modules and fields, together with the interaction among them and different mathematical branches and utilized features.

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Denote by pt I ∧ the constant functor from I op to Set, deﬁned by pt I ∧ (i) = {pt} for all i ∈ I . Note that pt I ∧ is a terminal object of I ∧ . We deﬁne a set, called the projective limit of β, by lim β = Hom I ∧ (pt I ∧ , β) . 2) lim β ←− i {xi }i ∈ β(i), i∈I , β(i), and it is immediately checked that: β(i) ; β(s)(x j ) = xi for all s ∈ Hom I (i, j) . i Since I and β(i) are small, lim β is a small set. The next result is obvious. 1. Let β : I op − → Set be a functor and let X ∈ Set. There is a natural isomorphism ∼ → lim Hom Set (X, β) , Hom Set (X, lim β) − ←− ←− → Set, i → Hom Set (X, β(i)).

7. 12) Y G X0 X1 G Z. 12) is co-Cartesian if X 0 Y X 1 − → Z. 12) is Cartesian if Y −→ X 0 × Z X 1 . Assume that C admits ﬁnite coproducts. 2): Y ⇒ X0 X1 − →Z. Assume that C admits ﬁnite products. 12) is Cartesian if and only if the sequence below is exact: Y − → X0 × X1 ⇒ Z . 8. Let f : X − → Y be a morphism in a category C. (i) Assume that C admits ﬁber coproducts and denote by i 1 , i 2 : Y ⇒ Y X Y → Y (or simply σ ) the natural the coprojections. We denote by σY : Y X Y − → Y , that is, σY ◦ i 1 = σY ◦ i 2 = idY .

Let I be a category and assume that C admits inductive limits indexed by I . , (i) If A : C op − lim A(X i ) for any inductive system {X i }i∈I in C), then C A A(lim X i ) ←− −→ i∈I i∈I admits inductive limits indexed by I and j A : C A − → C commutes with such limits. (ii) If a functor F : C − → C commutes with inductive limits indexed by I , →C then for any Y ∈ C , CY admits inductive limits indexed by I and CY − commutes with such limits. Proof. (i) Let {(X i , u i )}i∈I be an inductive system in C A indexed by I .