nLab
satellite

If $T:X\to Y$ is an additive functor between abelian categories with sufficiently many projectives and injectives, then one defines its right satellite $S^{1}T$ and left satellite $S^{-1}T=S_{1}T:C\to D$ via the formulas

$$S^{1}T(A)=\ker (T(M)\to T(P))$$S^1 T (A) = ker(T(M)\to T(P))

$$S_{1}T(A)=\mathrm{coker}(T(Q)\to T(N))$$S_1 T (A) = coker(T(Q)\to T(N))

where $0\to M\to P\to A\to 0$ and $0\to A\to Q\to N\to 0$ are short exact sequences where $P$ is projective and $Q$ is injective in $X$. This definition does not depend on the choice of these exact sequences, and moreover $S^1$ and $S_1$ extend to functors.

Higher satellites are defined by $S^{n}T=(S^1{)}^{n}T$ and $S^{-n}T=S_{n}T=(S_1{)}^{n}T$ for $n\ge 0$. For every exact sequence $0\to A\to A\prime \to A″\to 0$ there are natural connecting morphisms such that $S^{n}T$ (with $-\mathrm{\infty }<n<\mathrm{\infty }$) evaluated at $A,A\prime ,A″$ compose a long exact sequence.

If $T$ is right exact then $S^{n}T=0$ for all $n>0$ and if $T$ is left exact then $S^{n}T=0$ for all $n<0$. If $T$ is covariant and $A$ projective then $(S^{n}T)(A)=0$ for $n<0$, for contravariant $T$ or injective $A$ replace $n<0$ with $n>0$ in the conclusion.

There is also an axiomatic definition of satellites and their relation to derived functors in the case when $T$ is half exact. See

• H. Cartan, S. Eilenberg, Homological algebra, Princeton UP 1953, chapter 3.

There are generalizations to non-additive categories. See

• G. Z. Janelidze, On satellites in arbitrary categories, Bull. Georgian Acad. Sci. 82 (1976), no. 3, 529-532, in Russian, with a reprint translated in English at arXiv:0809.1504.

for early work; and also in a bit different setup recent

• A. Rosenberg, Homological algebra of noncommutative ‘spaces’ I, 199 pages, preprint Max Planck, Bonn: MPIM2008-91.