By Masaki Kashiwara, Pierre Schapira (auth.)

Categories and sheaves, which emerged in the midst of the final century as an enrichment for the recommendations of units and features, look virtually all over the place in arithmetic nowadays.

This e-book 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 newest ends up in the literature, and occasionally beyond.

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

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**Extra resources for Categories and Sheaves**

**Example text**

Ii) Assume that C admits ﬁber products and denote by p1 , p2 : X ×Y X ⇒ X → X ×Y X (or simply δ) the natural the projections. We denote by δ X : X − → X , that is, p1 ◦ δ X = p2 ◦ δ X = id X . We morphism associated with id X : X − call δ X the diagonal morphism. Consider a category C which admits ﬁnite products and let X ∈ C. 13) as follows. For I ∈ Set f , we set I X (I ) := X (in particular, X (∅) = ptC ) , and for ( f : J − → I ) ∈ Mor(C), X ( f ): X I − →X J is the morphism whose composition with the j-th projection X J − → X is the → X .

Using Zorn’s Lemma, for each X ∈ C, choose Y ∈ C0 and an isomor∼ phism ϕ X : Y − → F(X ), and set F0 (X ) = Y . If f : X − → X is a morphism in F( f ) ∼ → F (X ) as the composition F (X ) − → F(X ) −−→ C, deﬁne F ( f ) : F (X ) − 0 0 0 0 ϕX ∼− F (X ). The fact that F commutes with the composition of morF(X ) ← 0 0 ϕX phisms is visualized by F(X ) y ϕX F( f ) G F(X ) y ϕX ∼ Y = F0 (X ) F0 ( f ) G F(X ) y ϕX ∼ G Y = F0 (X ) The other assertions are obvious. F(g) F0 (g) ∼ G Y = F0 (X ) . d. 12. Let C be a category.

I op ) to C, lim α (resp. lim α) is −→ ←− representable, we say that C admits inductive (resp. projective ) limits indexed by I . (iv) We say that a category C admits ﬁnite (resp. small ) projective limits if it admits projective limits indexed by ﬁnite (resp. small ) categories, and similarly, replacing “projective limits” with “inductive limits”. 1. 3. The deﬁnitions of C ∧ and C ∨ depend on the choice of the universe U. However, given a functor α : I − → C, the fact that lim α is representable −→ as well as its representative does not depend on the choice of the universe U such that I is U-small and C is a U-category, and similarly for projective limits.