The concept of a $\delta$ -functor is used to formalize the procedure of constructing long exact sequences from short exact sequences. Examples include derived functors and cohomology.

Definition.  Let $\mathcal{A}$ and $\mathcal{B}$ be Abelian categories. A delta functor ($\delta$ -functor) from $\mathcal{A}$ to $\mathcal{B}$ consists of a family of covariant additive functors $$ F^n\colon\mathcal{A}\to\mathcal{B}\quad(n=0,1,2,\ldots) $$ and for each exact sequence $$ 0\to A\to B\to C\to 0 $$ of objects in $\mathcal{A}$ a family of homomorphisms $$ \delta^n\colon F^n(C)\to F^{n+1}(A)\quad(n=0,1,2,\ldots) $$ such that the following two conditions hold:

1. For any exact sequence $0\to A\to B\to C\to 0$ as above, there is a corresponding long exact sequence $$ \vcenter{\openup\jot \ialign{\hfil$\displaystyle{#{}}$&$\displaystyle{#}$\hfil\crcr 0\to& F^0(A)\to F^0(B)\to F^0(C)\stackrel{\delta^0}{\longrightarrow}\cr &F^1(A)\to F^1(B)\to F^1(C)\stackrel{\delta^1}{\longrightarrow}\cdots.\cr}} $$
2. For any morphism between exact sequences
   
   
   
   and all integers $n\ge 0$ the diagram
   
   
   
   is commutative.