By Masaki Kashiwara

Different types and sheaves, which emerged in the midst of the final century as an enrichment for the options of units and capabilities, seem nearly all over the place in arithmetic nowadays.

This publication covers different types, homological algebra and sheaves in a scientific and exhaustive demeanour ranging from scratch, and maintains with complete proofs to an exposition of the latest ends up in the literature, and infrequently beyond.

The authors current the final conception of different types and functors, emphasising inductive and projective limits, tensor different types, representable functors, ind-objects and localization. Then they examine homological algebra together with additive, abelian, triangulated different types and in addition unbounded derived different types utilizing transfinite induction and obtainable gadgets. ultimately, sheaf concept in addition to twisted sheaves and stacks seem within the framework of Grothendieck topologies.

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**Extra info for Categories and Sheaves**

**Sample text**

They are often called the “Yoneda functors”. Hence CU∧ = Fct(C op , U-Set) , CU∨ = Fct(C op , U-Setop ) Fct(C, U-Set)op . Note that CU∧ and CU∨ are not U-categories in general. If C is U-small, then CU∧ and CU∨ are U-categories. In the sequel, we shall write C ∧ and C ∨ for short. 1) C∨ C op∧op and C ∨ is the opposite big category to the category of functors from C to Set. Hence, for X ∈ C, kC (X ) = (hC op (X op ))op . The next result, although it is elementary, is crucial for the understanding of the rest of the book.

I) Let F : C − → C be a functor. We say that F is half-full if for any pair of objects X, Y ∈ C such that F(X ) and F(Y ) are isomorphic in C , there exists an isomorphism X Y in C. ) (ii) We say that a subcategory C0 of C is half-full if the embedding functor is half-full. 18. Let F : C − → C be a faithful and half-full functor. Then there exists a subcategory C0 of C such that F(Ob(C)) ⊂ Ob(C0 ), F(Mor(C)) ⊂ C0 . Moreover, the Mor(C0 ) and F induces an equivalence of categories C → C is faithful and half-full.

I) The functor ϕ∗ ∈ Fct Fct(I, C), Fct(J, C) is deﬁned by ϕ∗ α = α ◦ ϕ for α ∈ Fct(I, C) . (ii) If the functor ϕ∗ admits a left adjoint, we denote it by ϕ † . 1) Hom Fct(I,C) (ϕ † β, α) Hom Fct(J,C) (β, ϕ∗ α) . 3 Kan Extension of Functors 51 (iii) If the functor ϕ∗ admits a right adjoint, we denote it by ϕ ‡ . 2) Hom Fct(I,C) (α, ϕ ‡ β) Hom Fct(J,C) (ϕ∗ α, β) . These functors of big categories are visualized by the diagram ϕ‡ ϕ∗ o Fct(I, C) o G Fct(J, C). ϕ† We have the adjunction morphisms id − → ϕ∗ ◦ ϕ † , ϕ∗ ◦ ϕ ‡ − → id .