By Eduard Casas-Alvero, Gerald E. Welters, Sebastian Xambo-Descamps
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Extra info for Algebraic Geometry. Proc. conf. Sitges (Barcelona), 1983
It is now easy to verify that the arrows ( B , g , B , Uf Y ) and (Y,f,Bg Uf Y ) satisfy the universal property; the space B, Uf Y (or any space homeomorphic to it) is the pushout space of f and g. A case of particular importance is when g is the inclusion of a closed subspace A into Y : then, we denote g by i : A + Y and the pushout space just by B Uf Y ; the space B Uf Y is the adjuntion of Y to B via f. The map f:Y+BufY obtained in the construction of the adjunction space B Uf Y and - in view of the universal property for pushouts - the compositions of f with any homeomorphism B Uf Y 2 2 are called characteristic maps of the adjunction.
The following statements are equivalent: 1) for any two given maps f : X x ( 0 ) -+ Z and G : A x I -+ Z which coincide when restricted to A x ( 0 ) there is a map F : X x I -+ Z such that F restricted to X x (0) is f and F restricted to A x I is G; 2) the space X = X x (0) U A x I is a retract of X x I ; 3) X is a strong deformation retract of X x I . 3. COFIBRATIONS 51 io denotes the inclusion of (0) into I . Then X x I is a weak pushout of these two arrows 2 is a retract of X x I . 2) j 3): Let T : X x I +X be a retraction; for every (2,t ) E X X I , write and notice that r,y(z,O) = z , r ~ ( z , O= ) 0 for every z E X and, for every a E A , ~ .
For every a : I -+ B such that a(0) = b, there exists a unique path a' : I + E such that a'(0) = e, and pa' = a. 1 Proof - Let U be an open covering of B satisfying the condition spelled out in the definition of covering map. Using the Lebesgue number of the covering a-'(U) of I , we can construct a subdivision 0 = t,, < tl < t 2 < ... 4, for every i = 0, ,n. Set a'(0) = e, and suppose that we have defined a' for every t E Assume [0, t i ] ;we are going to define a' in the closed interval [ti, -- that a([ti,tj+l]) c U E U and that p-'(U) is the disjoint union of the open sets V, of E , indexed by a set A.
Algebraic Geometry. Proc. conf. Sitges (Barcelona), 1983 by Eduard Casas-Alvero, Gerald E. Welters, Sebastian Xambo-Descamps