S^1 T (A) = ker(T(M)\to T(P))
S_1 T (A) = coker(T(Q)\to T(N))
where and are short exact sequences where is projective and is injective in .
This definition does not depend on the choice of these exact sequences, and moreover and extend to functors.
Higher satellites are defined by and for . For every exact sequence there are natural connecting morphisms such that (with ) evaluated at compose a long exact sequence.
If is right exact then for all and if is left exact then for all . If is covariant and projective then for , for contravariant or injective replace with in the conclusion.
There is also an axiomatic definition of satellites and their relation to derived functors in the case when is half exact. See
There are generalizations to non-additive categories. See
for early work; and also in a bit different setup recent
or for a shorter presentation, Ch. 3 or so of the survey