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By M. Suzuki
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Additional resources for Group Theory I (Grundlehren der mathematischen Wissenschaften)
If A ⊂ X is a subset, then there is an equalizer G X A p ∗ GG X/A. The same holds for subobjects A ⊂ X of presheaves, and hence for subobjects of sheaves, since the associated sheaf functor L2 preserves finite limits. Statement 3) follows. For statement 4), observe that the map θ appears in an equalizer θ F G G f g GG K since θ is a monomorphism. But θ is an epimorphism, so f = g. But then 1G : G → G factors through θ , giving a section σ : G → F . Finally, θσ θ = θ and θ is a monomorphism, so σ θ = 1.
3 Geometric Morphisms Suppose that C and D are Grothendieck sites. A geometric morphism f : Shv(C) → Shv(D) consists of functors f∗ : Shv(C) → Shv(D) and f ∗ : Shv(D) → Shv(C) such that f ∗ is left adjoint to f∗ and f ∗ preserves finite limits. 3 Geometric Morphisms 43 The left adjoint f ∗ is called the inverse image functor, while f∗ is called the direct image . The inverse image functor f ∗ is left and right exact in the sense that it preserves all finite colimits and limits, respectively. e.
The constant sheaf construction still picks out global sections of sheaves F , by adjointness. There is a natural bijection hom(L2 (∗), F ) ∼ = Γ∗ (F ) relating sheaf morphisms L2 (∗) → F with elements of the inverse limit Γ∗ (F ) = lim F (U ). ← − U ∈C 38 3 Some Topos Theory For example, if F is a sheaf on the étale site et|S , then there is an identification Γ∗ F ∼ = F (S) (note the standard abuse of notation), since the identity map S → S is terminal in et|S . 14 1) The associated sheaf functor preserves all finite limits.
Group Theory I (Grundlehren der mathematischen Wissenschaften) by M. Suzuki