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Fk(H)1949 3903 y(H)2032 3944 y(H)2115 3986 y(H)2192 4025 y(H)-83 b(j)2133 3208 y Fh(1)p Fi(\012)p Fj(\023)p Fh(\()p Fj(X)q(;X)5 b Fh(\))1866 3293 y Fk(\010)1949 3252 y(\010)2032 3210 y(\010)2115 3169 y(\010)2192 3130 y(\010)-83 b(*)1055 3942 y Fh(db)12 b Fi(\012)p Fh(1)p Fi(\012)p Fh(1)1183 4025 y Fk(\010)1266 3983 y(\010)1349 3942 y(\010)1432 3900 y(\010)1510 3861 y(\010)-83 b(*)0 4313 y Fq(comm)m(utes.)43 b(The)34 b(exterior)e(p)s(ortion)f(of)i(this) f(diagram)e(yields)1168 4527 y Fm(!)t Fq(\()p Fm(X)8 b Fq(\))296 b Fm(!)t Fq(\()p Fm(X)8 b Fq(\))21 b Fl(\012)i Fq(co)s(end\()p Fm(!)t Fq(\))p 1426 4503 239 4 v 1582 4501 a Fk(-)1469 4466 y Fj(\016)r Fh(\()p Fj(X)5 b Fh(\))1824 4917 y Fm(!)t Fq(\()p Fm(X)j Fq(\))21 b Fl(\012)i Fm(M)5 b(:)p 2061 4824 4 254 v 2063 4824 a Fk(?)2102 4714 y Fh(1)p Fi(\012)k Fe(e)-48 b Fj(')1446 4713 y(')p Fh(\()p Fj(X)5 b Fh(\))1419 4612 y Fk(H)1502 4653 y(H)1585 4695 y(H)1668 4736 y(H)1751 4778 y(H)1834 4819 y(H)1843 4823 y(H)-83 b(j)149 5099 y Fq(It)42 b(remains)f(to)h(sho)m(w)h(that)54 b Ff(e)-67 b Fm(')43 b Fq(:)h(co)s(end)q(\()p Fm(!)t Fq(\))f Fl(\000)-60 b(!)43 b Fm(M)53 b Fq(is)41 b(uniquely)h (determined.)72 b(Let)54 b Ff(e)-67 b Fm(')3490 5114 y Fh(0)3573 5099 y Fq(:)0 5216 y(co)s(end)q(\()p Fm(!)t Fq(\))42 b Fl(\000)-60 b(!)43 b Fm(M)53 b Fq(b)s(e)42 b(another)g(morphism)e(with)h Fm(')p Fq(\()p Fm(X)8 b Fq(\))43 b(=)g(\(1)28 b Fl(\012)41 b Ff(e)-67 b Fm(')2618 5231 y Fh(0)2657 5216 y Fq(\))p Fm(\016)t Fq(\()p Fm(X)8 b Fq(\))42 b(for)f(all)e Fm(X)52 b Fl(2)43 b(D)s Fq(.)p eop %%Page: 97 7 97 6 bop 1240 236 a Fo(4.)35 b(FINITE)e(RECONSTR)n(UCTION)1136 b(97)0 425 y Fq(Then)34 b(the)f(follo)m(wing)d(diagram)g(comm)m(utes) 667 604 y Fm(!)t Fq(\()p Fm(X)8 b Fq(\))897 568 y Fi(\003)958 604 y Fl(\012)22 b Fm(!)t Fq(\()p Fm(X)8 b Fq(\))1259 b(co)s(end)q(\()p Fm(!)t Fq(\))p 1315 581 1195 4 v 2427 579 a Fk(-)1801 543 y Fj(\023)p Fh(\()p Fj(X)q(;X)5 b Fh(\))1290 897 y Fm(!)t Fq(\()p Fm(X)j Fq(\))1520 861 y Fi(\003)1581 897 y Fl(\012)22 b Fm(!)t Fq(\()p Fm(X)8 b Fq(\))22 b Fl(\012)g Fq(co)s(end)q(\()p Fm(!)t Fq(\))1480 742 y Fh(1)p Fi(\012)p Fj(\016)r Fh(\()p Fj(X)5 b Fh(\))1181 675 y Fk(P)1264 703 y(P)1347 731 y(P)1430 758 y(P)1513 786 y(P)1566 804 y(P)-83 b(q)1101 1035 y Fh(1)p Fi(\012)p Fj(')p Fh(\()p Fj(X)5 b Fh(\))1045 731 y Fk(@)1128 814 y(@)1211 897 y(@)1294 980 y(@)1377 1063 y(@)1460 1146 y(@)1543 1229 y(@)1626 1312 y(@)1703 1388 y(@)-83 b(R)2057 744 y Fh(ev)13 b Fi(\012)p Fh(1)2059 804 y Fk(\020)2142 776 y(\020)2225 748 y(\020)2308 721 y(\020)2391 693 y(\020)2444 675 y(\020)-83 b(1)p 1852 1389 4 449 v 1854 1389 a(?)1893 1182 y Fh(1)p Fi(\012)p Fh(1)p Fi(\012)9 b Fe(e)-48 b Fj(')2119 1191 y Fa(0)p 2730 1389 4 741 v 2732 1389 a Fk(?)2780 1036 y Fe(e)g Fj(')2817 1045 y Fa(0)1431 1482 y Fm(!)t Fq(\()p Fm(X)8 b Fq(\))1661 1446 y Fi(\003)1722 1482 y Fl(\012)22 b Fm(!)t Fq(\()p Fm(X)8 b Fq(\))22 b Fl(\012)g Fm(M)402 b(M)5 b(;)p 2306 1459 335 4 v 2558 1457 a Fk(-)2386 1431 y Fh(ev)14 b Fi(\012)p Fh(1)0 1627 y Fq(hence)34 b(w)m(e)g(ha)m(v)m(e)46 b Ff(e)-68 b Fm(')703 1642 y Fh(0)771 1627 y Fq(=)39 b Ff(e)-67 b Fm(')p Fq(.)p 3529 1627 4 66 v 3533 1565 59 4 v 3533 1627 V 3591 1627 4 66 v 149 1802 a Fn(Corollary)36 b(3.4.4.)50 b Fg(The)44 b(functor)h Fq(Nat\()p Fm(!)t(;)17 b(!)32 b Fl(\012)e Fm(M)10 b Fq(\))46 b Fg(is)f(a)g(r)-5 b(epr)g(esentable)44 b(functor)h(in)g Fm(M)0 1918 y Fg(r)-5 b(epr)g(esente)g(d)34 b(by)h Fq(co)s(end)q(\()p Fm(!)t Fq(\))p Fg(.)149 2092 y Fc(Pr)n(oof.)42 b Fq(The)34 b(univ)m(ersal)e(problem)g(implies)e(the) j(isomorphism)1000 2249 y(Nat\()p Fm(!)t(;)17 b(!)24 b Fl(\012)f Fm(M)10 b Fq(\))1662 2222 y Fl(\030)1663 2253 y Fq(=)1768 2249 y(Mor)1944 2264 y Fi(C)1989 2249 y Fq(\(co)s(end\()p Fm(!)t Fq(\))p Fm(;)17 b(M)10 b Fq(\))0 2406 y(and)40 b(the)h(univ)m(ersal)f(natural)e(transformation)g Fm(\016)45 b Fq(:)40 b Fm(!)k Fl(\000)-60 b(!)40 b Fm(!)31 b Fl(\012)d Fq(co)s(end\()p Fm(!)t Fq(\))40 b(is)f(mapp)s(ed)h(to)g (the)0 2522 y(iden)m(tit)m(y)32 b(under)i(this)e(isomorphism.)p 3529 2522 V 3533 2460 59 4 v 3533 2522 V 3591 2522 4 66 v 149 2688 a(It)h(is)f(also)g(p)s(ossible)g(to)g(construct)i(an)e (isomorphism)908 2845 y(Nat\()p Fm(!)t(;)17 b(!)1280 2804 y Fi(0)1323 2845 y Fl(\012)23 b Fm(M)10 b Fq(\))1593 2818 y Fl(\030)1594 2850 y Fq(=)1699 2845 y(Mor)1875 2860 y Fi(C)1920 2845 y Fq(\(cohom)o(\()p Fm(!)2337 2804 y Fi(0)2360 2845 y Fm(;)17 b(!)t Fq(\))p Fm(;)g(M)10 b Fq(\))0 3002 y(for)22 b(di\013eren)m(t)g(functors)h Fm(!)t(;)17 b(!)1054 2966 y Fi(0)1103 3002 y Fq(:)28 b Fl(D)i(\000)-60 b(!)28 b(C)h Fq(and)22 b(th)m(us)h(de\014ne)h Fg(c)-5 b(ohomomorphism)23 b(obje)-5 b(cts)p Fq(.)39 b(Observ)m(e)0 3119 y(that)32 b(only)f Fm(!)489 3082 y Fi(0)543 3119 y Fq(has)h(to)g(tak)m(e)g(v)-5 b(alues)32 b(in)f Fl(C)1500 3134 y Fh(0)1572 3119 y Fq(since)h(then)g(w)m(e)h(can) f(build)f(ob)5 b(jects)33 b Fm(!)2999 3082 y Fi(0)3021 3119 y Fq(\()p Fm(X)8 b Fq(\))3186 3082 y Fi(\003)3246 3119 y Fl(\012)21 b Fm(!)t Fq(\()p Fm(X)8 b Fq(\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF