Study Of The Problems Of Homotopy Theory In Category Of Morphisms

In this paper,we study the construction of the group of the under space self-homotopy equivalence of the objective sums in the under space category which is dual to fibrewise category,and in the category MAP,we continue to study the characterization of fibrations and the induced fibrations.we obtain the following main results:Theorem A In category C,if the elements of Aut(X + Y) are all reduced and if Aut^X (X + Y) and Aut^Y(X + Y) are subgroups of Aut(X + Y),thenAut(X + Y) = Aut^Y(X + Y)Aut^X(X + Y).Theorem B In category HNCW^A,if the elements of Aut(X/AVY/A) are all diagonalized,then Aut(s +^A t) - Aut{ix)Aut(i_Y)Theorem C M-fibrewise map (φ,α ) : (E₁,p₁,B₁)→ (E₂,P₂,B₂) is a M-fibrewise fibration if and only if there exists a M-fibrewise lifting function of (φ,α).Theorem D If (φ,α) : (E₁,p₁,B₁) →{E₂,P₂,B₂) is a M-fibrewise cofibration (E₁,E₂,B₁,B₂ are all locally compacted Hausdorff spaces),then the M-fibrewise map induced by continued map f : X →Y is a M-fibrewise fibration.