nLab
delta-functor

Definition

Definition

Let $𝒜$, $ℬ$ be two abelian categories.

A homological $\delta$-functor from $𝒜$ to $ℬ$ is for each $n\in \mathbb{N}$ a functor

$$T_n:𝒜\to ℬ$$T_n : \mathcal{A} \to \mathcal{B}

equipped for each short exact sequence $0\to A\to B\to C\to 0$ in $𝒜$ with a natural transformation

$${\delta }_n:T_n(C)\to T_{n-1}(A)$$\delta_n : T_n(C) \to T_{n-1}(A)

such that

for each such short exact sequence there is, naturally a long exact sequence

$$\cdots T_{n+1}(C)\stackrel{\delta }{\to }{T}_n(A)\to T_n(B)\to T_n(C)\stackrel{\delta }{\to }{T}_{n-1}(A)\to \cdots .$$\cdots T_{n+1}(C) \stackrel{\delta}{\to} T_n(A) \to T_n(B) \to T_n(C) \stackrel{\delta }{\to} T_{n-1}(A) \to \cdots \,.

Examples

The archetypical example is the chain homology functor

$$H_{•}(-):{\mathrm{Ch}}_{•}(𝒜)\to 𝒜$$H_\bullet(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

from the category of chain complexes of some abelian category (for $\mathbb{N}$-graded complexes).

The universal example are (non-total) right derived functors.

References

The notion is due to

• Alexander Grothendieck, Sur quelques points d’algèbre homologique

A textbook account is for instance section 2.1 of

• Charles Weibel, An Introduction to Homological Algebra