Introduction to Categories (Work in Progress) (original) (raw)
Definition 2. Let C be a category and A, B ∈ Obj (C). We say A and B are isomorphic if there exist f ∈ Hom C (A, B) and g ∈ Hom C (B, A) such that g ○ f = 1 A and f ○ g = 1 B. We write A ≅ B and call f and g isomorphisms. Definition 3. Let C be a category. A morphism f of C is called monic or a mono(morphism) (resp. epic or an epi(morphism)) if f ○ g = f ○ h implies g = h (resp. g ○ f = h ○ f implies g = h) Definition 4. C is a small category if both Obj (C) and Hom (C) are sets and not proper classes, and large otherwise. It is locally small if for every A, B ∈ Obj (C), Hom C (A, B) is a set (called a homset). Definition 5. A concrete category C is (for the time being) one whose objects are sets, potentially with additional structures (i.e. have underlying sets) and whose morphisms are maps between these sets. Definition 6. Let C and C ′ be two categories. We say C ′ is a subcategory of C if-Obj (C ′) ⊂ Obj (C).-For every A ′ , B ′ ∈ Obj (C ′), Hom C ′ (A ′ , B ′) ⊂ Hom C (A ′ , B ′).-morphism composition in C ′ coincides with that in C.-for every A ′ ∈ Obj (C ′), 1 A ′ in Hom C (A ′ , A ′) is equal to 1 A ′ in Hom C ′ (A ′ , A ′). Definition 7. We say a subcategory C ′ of C is full if for every A ′ , B ′ ∈ Obj (C ′) we have Hom C ′ (A ′ , B ′) = Hom C (A ′ , B ′).
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