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1 1 1 0 bop 0 241 a Fg(Mathematisc)m(hes)37 b(Institut)2314 b(SS)38 b(2004)0 362 y(der)g(Univ)m(ersit)649 367 y(\177)650 362 y(at)d(M)892 367 y(\177)889 362 y(unc)m(hen)2319 b(Blatt)36 b(8)0 482 y(Prof.)h(Dr.)g(B.)g(P)m(areigis)1424 981 y Ff(Kategorien)l(theorie)70 1209 y Fe(29.)49 b(Zeigen)30 b(Sie:)g(ein)g(spaltender)h(Epimorphism)m(us)e(ist)h(ein)g(Epimorphism) m(us.)g(Wie)g(steh)m(t)h(es)g(um)f(die)244 1329 y(en)m(tsprec)m(hende) 36 b(Aussage)e(f)1284 1333 y(\177)1281 1329 y(ur)e(Monomorphismen?)70 1533 y(30.)49 b(Zeigen)41 b(Sie,)h(da\031)g(die)f(folgenden)h(Aussagen) h(f)2036 1537 y(\177)2033 1533 y(ur)f(einen)g(Morphism)m(us)h Fd(f)54 b Fe(:)44 b Fd(B)49 b Fc(\000)-60 b(!)44 b Fd(C)49 b Fe(in)41 b(einer)244 1653 y(Kategorie)31 b Fc(C)779 1657 y Fe(\177)779 1653 y(aquiv)-5 b(alen)m(t)32 b(sind:)285 1856 y(\(a\))48 b Fd(f)c Fe(ist)32 b(ein)g(Isomorphism)m(us.)280 2018 y(\(b\))48 b Fd(f)c Fe(ist)32 b(so)m(w)m(ohl)h(ein)f(Monomorphism) m(us)g(als)g(auc)m(h)h(ein)f(spaltender)h(Epimorphism)m(us.)290 2180 y(\(c\))49 b Fd(f)44 b Fe(ist)32 b(so)m(w)m(ohl)h(ein)f (Epimorphism)m(us)f(als)h(auc)m(h)h(ein)g(spaltender)g(Monomorphism)m (us.)70 2384 y(31.)90 b(\(a\))48 b(Ein)35 b(Homomorphism)m(us)f(v)m(on) i(Graphen)g(ist)e(genau)i(dann)g(ein)f(Monomorphism)m(us,)g(w)m(enn)458 2504 y(b)s(eide)e(Abbildungen)f Fd(f)1341 2519 y Fb(0)1408 2504 y Fe(:)c Fd(G)1540 2519 y Fb(0)1607 2504 y Fc(\000)-60 b(!)28 b Fd(H)1833 2519 y Fb(0)1905 2504 y Fe(und)33 b Fd(f)2148 2519 y Fb(1)2215 2504 y Fe(:)28 b Fd(G)2347 2519 y Fb(1)2414 2504 y Fc(\000)-60 b(!)27 b Fd(H)2639 2519 y Fb(1)2711 2504 y Fe(injektiv)32 b(sind)280 2666 y(\(b\))48 b(Ein)e(Homomorphism)m(us)e(v)m(on)i(Graphen)g(ist)f(genau)h (dann)g(ein)g(Epimorphism)m(us,)f(w)m(enn)458 2786 y(b)s(eide)33 b(Abbildungen)f Fd(f)1341 2801 y Fb(0)1408 2786 y Fe(:)c Fd(G)1540 2801 y Fb(0)1607 2786 y Fc(\000)-60 b(!)28 b Fd(H)1833 2801 y Fb(0)1905 2786 y Fe(und)33 b Fd(f)2148 2801 y Fb(1)2215 2786 y Fe(:)28 b Fd(G)2347 2801 y Fb(1)2414 2786 y Fc(\000)-60 b(!)27 b Fd(H)2639 2801 y Fb(1)2711 2786 y Fe(surjektiv)33 b(sind)70 2990 y(32.)49 b(Die)43 b(nat)577 2994 y(\177)574 2990 y(urlic)m(hen)h(Zahlen)f Fa(N)60 b Fe(und)44 b(die)g(ganzen)h(Zahlen)f Fa(Z)d Fe(sind)k(b)s(eides)f(Monoide)g(un)m(ter)i(der)244 3110 y(Addition.)21 b(Die)g(Inklusions-Abbildung)g Fd(\023)29 b Fe(:)e Fa(N)43 b Fc(\000)-60 b(!)28 b Fa(Z)19 b Fe(ist)j(ein)g (Homomorphism)m(us)f(v)m(on)i(Monoiden,)244 3230 y(der)33 b(sic)m(herlic)m(h)f(nic)m(h)m(t)h(surjektiv)g(ist.)f(Zeigen)g(Sie,)h (dass)g Fd(\023)g Fe(ein)f(Epimorphism)m(us)g(ist.)0 3459 y(Abgab)s(e:)h(F)-8 b(reitag,)31 b(18.6.2004,)g(15)h(Uhr,)h(in)f (der)h(V)-8 b(orlesung.)0 3579 y(Bitte)32 b(geb)s(en)h(Sie)f(auf)g (Ihrer)i(L)1152 3583 y(\177)1152 3579 y(osung)e(Ihren)i(Namen)e(an.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF