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  1. \documentclass{article}
  2. \usepackage{amsmath}
  3. \usepackage{amsthm}
  4. \usepackage{amssymb}
  5. \usepackage{amsfonts}
  6. \usepackage{url}
  7. \usepackage{xspace}
  8. \usepackage{xy}
  9. \usepackage{comment}
  10.  
  11.  
  12. \xyoption{all} \CompileMatrices
  13.  
  14.  
  15. \newcommand{\Zobr}[3]{{#1}\colon{#2}\to{#3}}
  16. \newcommand{\ol}[1]{\overline{#1}}
  17.  
  18.  
  19. %\addtolength{\hoffset}{-1cm} \addtolength{\textwidth}{2cm}
  20.  
  21. %\includecomment{mine}
  22. \excludecomment{mine}
  23.  
  24. \begin{document}
  25.  
  26. The fact that $P$ is [projective]%(http://en.wikipedia.org/wiki/Projective_object)
  27. means the following:
  28.  
  29. Whenever we have an epimorphism $\Zobr fXY$ and a morphism $\Zobr pPY$, then there exists a morphism $\Zobr{\ol p}PX$ such that $f\circ\ol p=p$.
  30.  
  31. $$
  32. \xymatrix@C=10pt@R=17pt{
  33. & {P} \ar@{-->}[ld]_{\overline{p}} \ar[rd]^{p} \\
  34. {X} \ar[rr]_{f} && {Y} }
  35. $$
  36.  
  37. Now we have the following situation
  38.  
  39. $$
  40. \xymatrix@C=10pt@R=17pt{
  41. & P \ar@<-3pt>[d]_{r} \\
  42. & A \ar[rd]^{p} \ar@<-3pt>[u]_{s} \\
  43. X \ar[rr]_f && Y
  44. }
  45. $$
  46.  
  47. The rest of solution is: draw the obvious arrows to complete the diagram.
  48.  
  49. This would be a logical place to stop, if you only want a hint and want to do the rest by yourself.
  50. Since it is already some time since you posted your question, I guess posting full solution will not do much harm.
  51. (And of course you can simply ignore the rest of you post.)
  52.  
  53. We have the morphism $q=p\circ r$. Since $P$ is projective, there is a morphism $\ol q$ such that $f\circ \ol q=q=p\circ r$. Thus we get
  54. $$f\circ\ol q \circ s=p\circ r\circ s=p\circ 1_A=p,$$
  55. i.e. we have shown that there is a morphism $\ol p = \ol q\circ s$ fulfilling $f\circ\ol p=p$.
  56.  
  57. $$
  58. \xymatrix@C=10pt@R=17pt{
  59. & P \ar@<-3pt>[d]_{r} \ar[rdd]^q \ar[ldd]_{\overline q} \\
  60. & A \ar[rd]_{p} \ar@<-3pt>[u]_{s} \ar[ld]^{\overline p} \\
  61. X \ar[rr]_f && Y
  62. }
  63. $$
  64.  
  65.  
  66.  
  67.  
  68.  
  69. \end{document}
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