By Randall R. Holmes

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**Extra resources for Category Theory [Lecture notes]**

**Example text**

Let U be an open subset of X. Since γπ = α, we have π −1 (γ −1 (U )) = α−1 (U ), which is open in B since α is continuous. Hence, γ −1 (U ) is open in Q. We conclude that γ is continuous and the argument is complete. 3 Example (Coequalizers exist in Grp) Let λ1 , λ2 : A → B be two group homomorphisms. Let N be the normal closure in B of the set S = {λ1 (a)λ2 (a)−1 | a ∈ A}. This means that N is the intersection of all normal subgroups of B containing S. Put Q = B/N and let π : B → Q be the canonical epimorphism.

Then πi ιγ = σi γ = αi and similarly πi ιγ = αi (i = 1, 2), so the uniqueness assumption in the definition of product gives ιγ = ιγ. 4, ι is monic, so γ = γ and the claim is established. (ii ⇒ i) Assume that pullbacks exist in C and that C has a terminal object t. To prove that finite products exist in C it suffices to show that a product exists for a family {a1 , a2 } of two objects of C (Exercise 3–2). Since 35 t is terminal, there exist morphisms λi : ai → t (i = 1, 2) in C. By assumption, there exists a pullback (p, (π1 , π2 )) of the pair (λ1 , λ2 ), that is, a terminal object of the corresponding auxiliary category Dpb .

For a finite-dimensional vector space V , define ηV : V → K dim V by ηV (v) = [v], where [v] is the coordinate vector of v relative to the chosen basis of V . We claim that η : 1D → F G is a natural isomorphism. Since ηV is an isomorphism for each V it suffices to check the naturality condition. Let α : V → V be a morphism in D and put n = dim V , n = dim V . We check commutativity of the diagram on the right: V α V ηV V α V / Kn ηV 58 F G(α) / Kn . For v ∈ V we have (ηV α)(v) = [α(v)] = Mα [v] = µMα ([v]) = (F G(α)ηV )(v), where the second equality uses the definition of Mα .

### Category Theory [Lecture notes] by Randall R. Holmes

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