PlanetMath: delta functor

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delta functor

(Definition)

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

$\begin{displaymath} F^n\colon\mathcal{A}\to\mathcal{B}\quad(n=0,1,2,\ldots) \end{displaymath}$

and for each exact sequence

$\begin{displaymath} 0\to A\to B\to C\to 0 \end{displaymath}$

of objects in $\mathcal{A}$ a family of homomorphisms

$\begin{displaymath} \delta^n\colon F^n(C)\to F^{n+1}(A)\quad(n=0,1,2,\ldots) \end{displaymath}$

such that the following two conditions hold:

For any exact sequence $0\to A\to B\to C\to 0$ as above, there is a corresponding long exact sequence

$\begin{displaymath} \vcenter{\openup\jot \ialign{\hfil$\displaystyle{ ...$
For any morphism between exact sequences

$\begin{displaymath}\begin{xy} *!C\xybox{ \xymatrix{ 0\ar[r] & A\ar[r]\ar[d] & B\... ... 0\ar[r] & A'\ar[r] & B'\ar[r] & C'\ar[r] & 0 \ } } \end{xy}\end{displaymath}$

and all integers $n\ge 0$ the diagram

$\begin{displaymath}\begin{xy} *!C\xybox{ \xymatrix{ F^n(C) \ar[r]^{\delta^n} \ar... ...d] \ F^n(C') \ar[r]^{\delta^n} & F^{n+1}(A') \ } } \end{xy}\end{displaymath}$

is commutative.

"delta functor" is owned by pbruin.

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See Also: derived functor

Other names:

$\delta$ -functor

Keywords:

exact sequence, homological algebra

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Cross-references: commutative, integers, morphism, homomorphisms, objects, additive functors, abelian categories, cohomology, derived functors, short exact sequences, exact sequences
There is 1 reference to this entry.

This is version 1 of delta functor, born on 2005-08-12.
Object id is 7317, canonical name is DeltaFunctor.
Accessed 1812 times total.

Classification:

AMS MSC:	18G10 (Category theory; homological algebra :: Homological algebra :: Resolutions; derived functors)
	18G99 (Category theory; homological algebra :: Homological algebra :: Miscellaneous)

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