\( \usepackage{mathrsfs} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\cHom}{\underline{Hom}} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\inj}{inj} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\coim}{coim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\eq}{eq} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\ob}{ob} \DeclareMathOperator{\coeq}{coeq} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\Ab}{\text{Ab}} \DeclareMathOperator{\Top}{\text{Top}} \DeclareMathOperator{\Rmod}{R\text{mod}} \DeclareMathOperator{\SSet}{\text{SSet}} \DeclareMathOperator{\sAb}{\text{sAb}} \DeclareMathOperator{\cA}{\mathcal{A}} \DeclareMathOperator{\cB}{\mathcal{B}} \DeclareMathOperator{\cC}{\mathcal{C}} \DeclareMathOperator{\cO}{\mathcal{O}} \DeclareMathOperator{\cM}{\mathcal{M}} \DeclareMathOperator{\sC}{\mathscr{C}} \DeclareMathOperator{\sF}{\mathscr{F}} \DeclareMathOperator{\sW}{\mathscr{W}} \DeclareMathOperator{\dd}{\partial} \DeclareMathOperator{\rd}{\mathrm{d}} % Rings \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\PP}{\mathbf{P}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \)

Section1.3Categorical notions

Subsection1.3.1Abelian Categories

Example1.3.1
\(\Rmod\) - the category of left \(R\)-modules for \(R\) an associative ring is an abelian category.
Example1.3.2
The categories of sheaves of abelian groups on a topological space, sheaves of \(\cO\)-modules on a scheme and (quasi-)coherent sheaves on a scheme are all abelian.
Definition1.3.3Additive categories
An additive category is a category in which:
  1. Every hom-space has the structure of an abelian group.
  2. There exists a 0-object (one with exactly one map to and from every other object).
  3. Finite products exist (these are automatically equal to sums \(A\times B = A \oplus B = A \amalg B\)).
In such a category we let \[\ker(f) = \eq(\xymatrix@+=2pc{A \ar@<0.5ex>[r]^f \ar@<-0.5ex>[r]_0 & B})\] and \[\coker(f) = \coeq(\xymatrix@+=2pc{A \ar@<0.5ex>[r]^f \ar@<-0.5ex>[r]_0 & B}).\]
Definition1.3.4Abelian categories
An abelian category\(\cA\) is an additive category in which:
  1. Every map \(f\) has a kernel and cokernel.
  2. For all \(f\) we have \(\coker(\ker(f)) = \im(f) = \coim(f) = \ker(\coker(f))\).
Example1.3.5
Let \(\cB\) be the category of pairs of vector spaces \(V\subset W\), with morphisms the compatible linear maps. Consider the natural map \(f\colon 0\subset V \to V\subset V\), we then have \(\im f \cong 0\subset V\) but \(\coim f \cong V\subset V\). So this category is not abelian.

From now on we take \(\cA\) to be any abelian category.

Subsection1.3.2Exactness

Definition1.3.6Exact sequences
A sequence of morphisms \[A\xrightarrow{f} B \xrightarrow{g}C\] in \(\cA\) is exact at \(B\) if \(\im f = \ker g\). A sequence is then exact if it is exact everywhere. An exact sequence of the form \[0\to A \to B \to C \to 0\] is called a short exact sequence.
Definition1.3.7Mono and epi morphisms
A morphism \(f\) is a monomorphism if \(fg = fh \implies g=h\) and it is an epimorphism if \(gf = hf \implies g=h\).
Example1.3.8
In \(\Ab\) the following are exact sequences: \begin{gather*} 0\to \ZZ/2 \to \ZZ/2 \oplus \ZZ/2 \to \ZZ/2 \to 0\\ 0\to \ZZ/2 \to \ZZ/4\to \ZZ/2 \to 0\\ 0\to \ZZ \xrightarrow{\cdot 3} \ZZ\to \ZZ/3 \to 0 \end{gather*}
Definition1.3.9Additive functors
A functor of additive categories is additive if it is a homomorphism on hom-sets.