Equivalence of coverings criterion
Let
Proof
By discussion in Category of coverings with basepoint, if
there exists a unique i m β‘ π 1 π = i m β‘ π 1 π and π β π’ π π ( π , π₯ 0 ) ( π , π ) . Moreover, the identities π β π’ π π ( π , π₯ 0 ) ( π , π ) and i d π = i d ( Λ π , Λ π₯ 0 ) are the only morphisms in i d π = i d ( Λ π β² , Λ π₯ β² 0 ) and π’ π π ( π , π₯ 0 ) ( π , π ) respectively. Therefore π’ π π ( π , π₯ 0 ) ( π , π ) and π π = i d π , hence π π = i d π and π are equivalent. π