A commutative square (1) of morphisms is said to have a lifting if there is a morphism λ: B
1 → A
2 such that λϕ
1 = α and ϕ
2λ = β
1
Let us assume that we are working in a fixed abelian category
. Therefore, ϕ
i will have a kernel “Ki
” and a cokernel “Ci
” for i = 1, 2. Let k : K
1 → K
2 and c: C1 → C
2 denote the canonical morphisms induced by α and β.
We shall construct a short exact sequence (s.e.s.)
2
using the data of (1). We shall prove that (1) has a lifting if and only if k = 0, c = 0, and (2) represents the zero class in Ext1(C
1, K
2). Furthermore, if (1) has one lifting, then the liftings will be in one-to-one correspondence with the elements of the set |Hom(G
1, K
2)|.