Geodesic
Obstructions to Liftings in Commutative Squares
Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 977-982

Voir la notice de l'article provenant de la source Cambridge University Press

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)|. 
Pressman, Irwin S. Obstructions to Liftings in Commutative Squares. Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 977-982. doi: 10.4153/CJM-1971-105-2
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