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5179 y Fj(0)1735 5216 y Fl(\002)c Fn(g)t Fr(\))e Fl(\016)h Fr(\()p Fn(f)33 b Fl(\002)23 b Fn(N)10 b Fr(\).)p eop %%Page: 13 5 13 4 bop 1348 236 a Fp(4.)77 b(TENSOR)32 b(PR)n(ODUCTS)1245 b(13)149 425 y Fo(Prop)s(osition)36 b(8.4.9.)49 b Fk(The)35 b(fol)5 b(lowing)33 b(de\014ne)h(c)-5 b(ovariant)34 b(functors)149 562 y Fr(1.)42 b Fk(-)22 b Fl(\012)g Fn(N)39 b Fr(:)27 b Fo(Mo)s(d)q Fk(-)p Fn(R)i Fl(\000)-58 b(!)28 b Fo(Ab)p Fk(;)149 678 y Fr(2.)42 b Fn(M)33 b Fl(\012)22 b Fk(-)28 b Fr(:)g Fn(R)q Fk(-)p Fo(Mo)s(d)g Fl(\000)-58 b(!)28 b Fo(Ab)p Fk(;)149 794 y Fr(3.)42 b Fk(-)22 b Fl(\012)g Fk(-)28 b Fr(:)g Fo(Mo)s(d)q Fk(-)o Fn(R)c Fl(\002)e Fn(R)q Fk(-)p Fo(Mo)s(d)29 b Fl(\000)-58 b(!)28 b Fo(Ab)p Fk(.)149 956 y Fh(Pr)n(oof.)42 b Fr(\()p Fn(f)34 b Fl(\002)23 b Fn(g)t Fr(\))f Fl(\016)g Fr(\()p Fn(f)1034 919 y Fj(0)1080 956 y Fl(\002)h Fn(g)1231 919 y Fj(0)1254 956 y Fr(\))29 b(=)g Fn(f)11 b(f)1544 919 y Fj(0)1589 956 y Fl(\002)23 b Fn(g)t(g)1791 919 y Fj(0)1847 956 y Fr(implies)31 b(\()p Fn(f)i Fl(\012)2375 971 y Fm(R)2456 956 y Fn(g)t Fr(\))22 b Fl(\016)g Fr(\()p Fn(f)2736 919 y Fj(0)2782 956 y Fl(\012)2859 971 y Fm(R)2940 956 y Fn(g)2991 919 y Fj(0)3013 956 y Fr(\))29 b(=)g Fn(f)11 b(f)3303 919 y Fj(0)3349 956 y Fl(\002)23 b Fn(g)t(g)3551 919 y Fj(0)3573 956 y Fr(.)0 1072 y(F)-8 b(urthermore)32 b(1)614 1087 y Fm(M)715 1072 y Fl(\002)22 b Fr(1)863 1087 y Fm(N)958 1072 y Fr(=)28 b(1)1111 1087 y Fm(M)7 b Fj(\002)p Fm(N)1340 1072 y Fr(implies)30 b(1)1720 1087 y Fm(M)1821 1072 y Fl(\012)1898 1087 y Fm(R)1978 1072 y Fr(1)2027 1087 y Fm(N)2122 1072 y Fr(=)e(1)2275 1087 y Fm(M)7 b Fj(\012)2405 1098 y Fd(R)2454 1087 y Fm(N)2522 1072 y Fr(.)p 3529 1072 4 66 v 3533 1009 59 4 v 3533 1072 V 3591 1072 4 66 v 149 1233 a Fo(De\014nition)37 b(8.4.10.)49 b Fr(Let)24 b Fn(R)q Fr(,)i Fn(S)j Fr(b)s(e)c(rings)e(and) h(let)f Fn(M)35 b Fr(b)s(e)24 b(a)g(left)f Fn(R)q Fr(-mo)s(dule)f(and)i (a)g(righ)m(t)f Fn(S)6 b Fr(-)0 1350 y(mo)s(dule.)41 b Fn(M)e Fr(is)27 b(called)g(an)h Fn(R)q Fr(-)p Fn(S)6 b Fr(-)p Fk(bimo)-5 b(dule)27 b Fr(if)g(\()p Fn(r)s(m)p Fr(\))p Fn(s)g Fr(=)h Fn(r)s Fr(\()p Fn(ms)p Fr(\).)41 b(W)-8 b(e)29 b(de\014ne)h(Hom)3053 1365 y Fm(R)p Fr(-)p Fm(S)3190 1350 y Fr(\()p Fn(:M)5 b(:;)17 b(:N)5 b(:)p Fr(\))0 1466 y(:=)28 b(Hom)334 1481 y Fm(R)392 1466 y Fr(\()p Fn(:M)5 b(;)17 b(:N)10 b Fr(\))22 b Fl(\\)h Fr(Hom)1067 1481 y Fm(S)1118 1466 y Fr(\()p Fn(M)5 b(:;)17 b(N)5 b(:)p Fr(\).)149 1627 y Fo(Remark)37 b(8.4.11.)50 b Fr(Let)34 b Fn(M)1211 1642 y Fm(S)1297 1627 y Fr(b)s(e)g(a)g(righ)m(t)g Fn(S)6 b Fr(-mo)s(dule)32 b(and)j(let)e Fn(R)25 b Fl(\002)f Fn(M)41 b Fl(\000)-60 b(!)31 b Fn(M)45 b Fr(a)34 b(map.)48 b Fn(M)0 1743 y Fr(is)32 b(an)h Fn(R)q Fr(-)p Fn(S)6 b Fr(-bimo)s(dule)29 b(if)i(and)i(only)f(if)149 1880 y(1.)42 b Fl(8)p Fn(r)31 b Fl(2)d Fn(R)g Fr(:)g(\()p Fn(M)39 b Fl(3)28 b Fn(m)g Fl(7!)f Fn(r)s(m)h Fl(2)g Fn(M)10 b Fr(\))28 b Fl(2)g Fr(Hom)1874 1895 y Fm(S)1925 1880 y Fr(\()p Fn(M)5 b(:;)17 b(M)5 b(:)p Fr(\),)149 1996 y(2.)42 b Fl(8)p Fn(r)m(;)17 b(r)454 1960 y Fj(0)505 1996 y Fl(2)28 b Fn(R)q(;)17 b(m)28 b Fl(2)g Fn(M)38 b Fr(:)28 b(\()p Fn(r)d Fr(+)d Fn(r)1364 1960 y Fj(0)1387 1996 y Fr(\))p Fn(m)28 b Fr(=)f Fn(r)s(m)22 b Fr(+)g Fn(r)1940 1960 y Fj(0)1963 1996 y Fn(m)p Fr(,)149 2112 y(3.)42 b Fl(8)p Fn(r)m(;)17 b(r)454 2076 y Fj(0)505 2112 y Fl(2)28 b Fn(R)q(;)17 b(m)28 b Fl(2)g Fn(M)38 b Fr(:)28 b(\()p Fn(r)s(r)1244 2076 y Fj(0)1267 2112 y Fr(\))p Fn(m)f Fr(=)h Fn(r)s Fr(\()p Fn(r)1653 2076 y Fj(0)1676 2112 y Fn(m)p Fr(\),)149 2229 y(4.)42 b Fl(8)p Fn(m)28 b Fl(2)g Fn(M)39 b Fr(:)27 b(1)p Fn(m)h Fr(=)g Fn(m:)149 2390 y Fo(Lemma)38 b(8.4.12.)49 b Fk(L)-5 b(et)1071 2405 y Fm(R)1129 2390 y Fn(M)1223 2405 y Fm(S)1299 2390 y Fk(and)1479 2405 y Fm(S)1529 2390 y Fn(N)1607 2405 y Fm(T)1687 2390 y Fk(b)g(e)25 b(bimo)-5 b(dules.)41 b(Then)2524 2405 y Fm(R)2581 2390 y Fr(\()p Fn(M)11 b Fl(\012)2801 2405 y Fm(S)2853 2390 y Fn(N)f Fr(\))2979 2405 y Fm(T)3059 2390 y Fk(is)25 b(a)g(bimo)-5 b(dule)0 2506 y(by)35 b Fn(r)s Fr(\()p Fn(m)22 b Fl(\012)h Fn(n)p Fr(\))28 b(:=)f Fn(r)s(m)22 b Fl(\012)h Fn(n)35 b Fk(and)f Fr(\()p Fn(m)23 b Fl(\012)f Fn(n)p Fr(\))p Fn(t)28 b Fr(:=)g Fn(m)23 b Fl(\012)f Fn(nt)p Fk(.)149 2668 y Fh(Pr)n(oof.)42 b Fr(Ob)m(viously)i(w)m(e)h(ha)m(v)m(e)g(2.-4.)76 b(F)-8 b(urthermore)44 b(\()p Fn(r)32 b Fl(\012)2424 2683 y Fm(S)2505 2668 y Fr(id)o(\)\()p Fn(m)e Fl(\012)h Fn(n)p Fr(\))46 b(=)h Fn(r)s(m)30 b Fl(\012)g Fn(n)47 b Fr(=)0 2784 y Fn(r)s Fr(\()p Fn(m)22 b Fl(\012)h Fn(n)p Fr(\))32 b(is)g(a)h(homomorphism.)p 3529 2784 V 3533 2721 59 4 v 3533 2784 V 3591 2784 4 66 v 149 2945 a Fo(Corollary)j(8.4.13.)50 b Fk(Given)30 b(bimo)-5 b(dules)1736 2960 y Fm(R)1794 2945 y Fn(M)1888 2960 y Fm(S)1939 2945 y Fk(,)2001 2960 y Fm(S)2052 2945 y Fn(N)2130 2960 y Fm(T)2185 2945 y Fk(,)2246 2960 y Fm(R)2304 2945 y Fn(M)2408 2909 y Fj(0)2398 2971 y Fm(S)2449 2945 y Fk(,)2511 2960 y Fm(S)2562 2945 y Fn(N)2650 2909 y Fj(0)2640 2971 y Fm(T)2726 2945 y Fk(and)30 b(homomorphisms)0 3062 y Fn(f)40 b Fl(2)29 b Fr(Hom)386 3077 y Fm(R)p Fk(-)q Fm(S)526 3062 y Fr(\()p Fn(:M)5 b(:;)17 b(:M)892 3025 y Fj(0)916 3062 y Fn(:)p Fr(\))36 b Fk(and)f Fn(g)d Fl(2)e Fr(Hom)1585 3077 y Fm(S)t Fk(-)o Fm(T)1721 3062 y Fr(\()p Fn(:N)5 b(:;)17 b(:N)2055 3025 y Fj(0)2079 3062 y Fn(:)p Fr(\))p Fk(.)47 b(Then)35 b(we)g(have)g Fn(f)f Fl(\012)3005 3077 y Fm(S)3078 3062 y Fn(g)f Fl(2)c Fr(Hom)3457 3077 y Fm(R)p Fk(-)p Fm(T)0 3178 y Fr(\()p Fn(:M)k Fl(\012)269 3193 y Fm(S)342 3178 y Fn(N)5 b(:;)17 b(:M)627 3142 y Fj(0)674 3178 y Fl(\012)751 3193 y Fm(S)824 3178 y Fn(N)912 3142 y Fj(0)936 3178 y Fn(:)p Fr(\))p Fn(:)149 3339 y Fh(Pr)n(oof.)42 b Fn(f)33 b Fl(\012)692 3354 y Fm(S)766 3339 y Fn(g)t Fr(\()p Fn(r)s(m)21 b Fl(\012)i Fn(nt)p Fr(\))28 b(=)f Fn(f)11 b Fr(\()p Fn(r)s(m)p Fr(\))22 b Fl(\012)h Fn(g)t Fr(\()p Fn(nt)p Fr(\))k(=)g Fn(r)s Fr(\()p Fn(f)33 b Fl(\012)2352 3354 y Fm(S)2425 3339 y Fn(g)t Fr(\)\()p Fn(m)22 b Fl(\012)h Fn(n)p Fr(\))p Fn(t:)p 3529 3339 V 3533 3277 59 4 v 3533 3339 V 3591 3339 4 66 v 149 3501 a 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Fn(m\025)22 b Fl(\012)h Fn(n)28 b Fr(=)f Fn(m)c Fl(\012)g Fn(\025n)k Fr(=)h Fn(m)22 b Fl(\012)h Fn(n\025)p Fr(.)44 b(Th)m(us)34 b(the)f(follo)m(wing)d(diagram)g(of)i(functors)h(comm)m(utes:)991 5223 y Fc(K)18 b Fr(-)5 b Fo(Mo)s(d)764 b Fc(K)18 b Fr(-)5 b Fo(Mo)s(d)q Fr(-)p Fc(K)17 b Fn(:)p 1358 5190 706 4 v 1981 5188 a Fi(-)1694 5250 y Fn(\023)761 4731 y Fc(K)h Fr(-)5 b Fo(Mo)s(d)23 b Fl(\002)g Fc(K)17 b Fr(-)6 b Fo(Mo)s(d)263 b Fc(K)18 b Fr(-)5 b Fo(Mo)s(d)q Fr(-)p Fc(K)45 b Fl(\002)23 b Fc(K)17 b Fr(-)6 b Fo(Mo)s(d)q Fr(-)o Fc(K)p 1587 4703 206 4 v 1710 4701 a Fi(-)1594 4673 y Fn(\023)23 b Fl(\002)f Fn(\023)p 1158 5120 4 351 v 1160 5120 a Fi(?)1023 4959 y Fj(\012)1078 4970 y Fa(K)p 2328 5120 V 2330 5120 a Fi(?)2369 4959 y Fj(\012)2424 4970 y Fa(K)p eop %%Page: 14 6 14 5 bop 0 236 a Fp(14)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fr(So)45 b(w)m(e)g(can)g(consider)h Fc(K)17 b Fr(-)p Fo(Mo)s(d)51 b Fr(as)45 b(a)f(\(prop)s(er\))h(sub)s(category)h(of)e Fc(K)17 b Fr(-)6 b Fo(Mo)s(d)q Fr(-)o 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b(of)f(the)h(Righ)m(t)f(Unit:)44 b Fn(\032)28 b Fr(:)g Fn(M)k Fl(\012)23 b Fc(K)1754 1201 y Fl(\030)1755 1233 y Fr(=)1859 1229 y Fn(M)10 b Fk(.)149 1345 y Fr(4.)42 b(Symmetry)32 b(La)m(w:)46 b Fn(\034)39 b Fr(:)28 b Fn(M)33 b Fl(\012)22 b Fn(N)1466 1318 y Fl(\030)1467 1349 y Fr(=)1572 1345 y Fn(N)32 b Fl(\012)23 b Fn(M)10 b Fk(.)149 1462 y Fr(5.)42 b(Existence)34 b(of)e(Inner)h(Hom-F)-8 b(unctors:)44 b(Hom\()p Fn(P)35 b Fl(\012)23 b Fn(M)5 b(;)17 b(N)10 b Fr(\))2488 1434 y Fl(\030)2489 1466 y Fr(=)2593 1462 y(Hom)o(\()p Fn(P)s(;)17 b Fr(Hom)o(\()p Fn(M)5 b(;)17 b(N)10 b Fr(\)\))p Fk(.)149 1651 y Fh(Pr)n(oof.)42 b Fr(W)-8 b(e)33 b(only)f(describ)s(e)i(the)f(corresp)s(onding)f (homomorphisms.)149 1767 y(1.)44 b(Use)33 b(\(8.4.45.\))43 b(to)32 b(de\014ne)i Fn(\013)q Fr(\(\()p Fn(m)22 b Fl(\012)g Fn(n)p Fr(\))h Fl(\012)f Fn(p)p Fr(\))28 b(:=)g Fn(m)22 b Fl(\012)h Fr(\()p Fn(n)f Fl(\012)h Fn(p)p Fr(\).)149 1883 y(2.)44 b(De\014ne)33 b Fn(\025)27 b Fr(:)h Fc(K)46 b Fl(\012)22 b Fn(M)39 b Fl(\000)-60 b(!)28 b Fn(M)43 b Fr(b)m(y)33 b Fn(\025)p Fr(\()p Fn(r)25 b Fl(\012)e Fn(m)p Fr(\))28 b(:=)f Fn(r)s(m)p Fr(.)149 1999 y(3.)44 b(De\014ne)33 b Fn(\032)28 b Fr(:)g Fn(M)k Fl(\012)23 b Fc(K)51 b Fl(\000)-60 b(!)28 b Fn(M)43 b Fr(b)m(y)34 b Fn(\032)p Fr(\()p Fn(m)22 b Fl(\012)h Fn(r)s Fr(\))k(:=)h Fn(mr)s Fr(.)149 2116 y(4.)44 b(De\014ne)33 b Fn(\034)11 b Fr(\()p Fn(m)23 b Fl(\012)f Fn(n)p Fr(\))28 b(:=)g Fn(n)22 b Fl(\012)h Fn(m)p Fr(.)149 2232 y(5.)83 b(F)-8 b(or)45 b Fn(f)61 b Fr(:)50 b Fn(P)45 b Fl(\012)32 b Fn(M)60 b Fl(\000)-60 b(!)51 b Fn(N)56 b Fr(de\014ne)47 b Fn(\036)p Fr(\()p Fn(f)11 b Fr(\))49 b(:)i Fn(P)63 b Fl(\000)-60 b(!)50 b Fr(Hom\()p Fn(M)5 b(;)17 b(N)10 b Fr(\))46 b(b)m(y)h Fn(\036)p Fr(\()p Fn(f)11 b Fr(\)\()p Fn(p)p Fr(\)\()p Fn(m)p Fr(\))49 b(:=)0 2348 y Fn(f)11 b Fr(\()p Fn(p)22 b Fl(\012)g Fn(m)p Fr(\).)p 3529 2348 4 66 v 3533 2286 59 4 v 3533 2348 V 3591 2348 4 66 v 149 2546 a(Usually)36 b(one)h(iden)m(ti\014es)g(threefold)f(tensor)h (pro)s(ducts)g(along)e(the)i(map)f Fn(\013)h Fr(so)g(that)f(w)m(e)i (use)0 2662 y Fn(M)29 b Fl(\012)18 b Fn(N)29 b Fl(\012)18 b Fn(P)41 b Fr(=)28 b(\()p Fn(M)g Fl(\012)19 b Fn(N)10 b Fr(\))18 b Fl(\012)g Fn(P)42 b Fr(=)27 b Fn(M)i Fl(\012)18 b Fr(\()p Fn(N)29 b Fl(\012)18 b Fn(P)c Fr(\).)42 b(F)-8 b(or)30 b(the)h(notion)e(of)h(a)h(monoidal)c(or)j(tensor)0 2779 y(category)-8 b(,)33 b(ho)m(w)m(ev)m(er,)i(this)d(natural)g (transformation)e(is)i(of)g(cen)m(tral)h(imp)s(ortance.)149 2968 y Fo(Problem)j(8.4.1.)49 b Fr(1.)44 b(Giv)m(e)32 b(an)g(explicit)g(pro)s(of)f(of)h Fn(M)h Fl(\012)23 b Fr(\()p Fn(X)30 b Fl(\010)22 b Fn(Y)g Fr(\))2725 2940 y Fl(\030)2726 2972 y Fr(=)2830 2968 y Fn(M)33 b Fl(\012)23 b Fn(X)30 b Fl(\010)23 b Fn(M)32 b Fl(\012)23 b Fn(Y)e Fr(.)149 3084 y(2.)42 b(Sho)m(w)28 b(that)g(for)e(ev)m(ery)k(\014nite)d (dimensional)e(v)m(ector)j(space)h Fn(V)49 b Fr(there)28 b(is)f(a)g(unique)h(elemen)m(t)0 3126 y Fe(P)105 3152 y Fm(n)105 3230 y(i)p Ff(=1)240 3200 y Fn(v)287 3215 y Fm(i)338 3200 y Fl(\012)22 b Fn(v)488 3164 y Fj(\003)484 3225 y Fm(i)555 3200 y Fl(2)28 b Fn(V)44 b Fl(\012)22 b Fn(V)928 3164 y Fj(\003)1000 3200 y Fr(suc)m(h)34 b(that)e(the)h (follo)m(wing)d(holds)1226 3409 y Fl(8)p Fn(v)i Fl(2)c Fn(V)49 b Fr(:)1712 3315 y Fe(X)1772 3525 y Fm(i)1873 3409 y Fn(v)1924 3368 y Fj(\003)1920 3434 y Fm(i)1963 3409 y Fr(\()p Fn(v)t Fr(\))p Fn(v)2137 3424 y Fm(i)2193 3409 y Fr(=)27 b Fn(v)t(:)0 3684 y Fr(\(Hin)m(t:)44 b(Use)35 b(an)e(isomorphism)e(End\()p Fn(V)22 b Fr(\))1553 3657 y Fl(\030)1554 3689 y Fr(=)1659 3684 y Fn(V)44 b Fl(\012)24 b Fn(V)1939 3648 y Fj(\003)2012 3684 y Fr(and)33 b(dual)g(bases)h Fl(f)p Fn(v)2773 3699 y Fm(i)2801 3684 y Fl(g)f Fr(of)g Fn(V)55 b Fr(and)33 b Fl(f)p Fn(v)3399 3648 y Fj(\003)3395 3709 y Fm(i)3438 3684 y Fl(g)g Fr(of)0 3801 y Fn(V)79 3764 y Fj(\003)118 3801 y Fr(.\))149 3917 y(3.)41 b(Sho)m(w)25 b(that)f(the)h(follo)m(wing)d(diagrams)g(\(coherence)k(diagrams)d(or)h (constrain)m(ts\))h(comm)m(ute)0 4033 y(in)32 b Fc(K)17 b Fr(-)p Fo(Mo)s(d)7 b Fr(:)20 4219 y(\(\()p Fn(A)22 b Fl(\012)h Fn(B)5 b Fr(\))22 b Fl(\012)h Fn(C)7 b Fr(\))22 b Fl(\012)g Fn(D)538 b Fr(\()p Fn(A)22 b Fl(\012)h Fr(\()p Fn(B)k Fl(\012)c Fn(C)7 b Fr(\)\))22 b Fl(\012)g Fn(D)p 879 4196 478 4 v 1274 4194 a Fi(-)921 4158 y Fm(\013)p Ff(\()p Fm(A;B)s(;C)5 b Ff(\))p Fj(\012)p Ff(1)2750 4219 y Fn(A)22 b Fl(\012)h Fr(\(\()p Fn(B)k Fl(\012)c Fn(C)7 b Fr(\))22 b Fl(\012)g Fn(D)s Fr(\))p 2244 4196 V 2639 4194 a Fi(-)2275 4158 y Fm(\013)p Ff(\()p Fm(A;B)s Fj(\012)p Fm(C)q(;D)r Ff(\))p 433 4516 4 254 v 435 4516 a Fi(?)474 4406 y Fm(\013)p Ff(\()p Fm(A)p Fj(\012)p Fm(B)s(;C)q(;D)r Ff(\))p 3163 4516 V 3165 4516 a Fi(?)2729 4406 y Ff(1)p Fj(\012)p Fm(\013)p Ff(\()p Fm(B)s(;C)q(;D)r Ff(\))20 4609 y Fr(\()p Fn(A)g Fl(\012)h Fn(B)5 b Fr(\))22 b Fl(\012)h Fr(\()p Fn(C)29 b Fl(\012)22 b Fn(D)s Fr(\))1900 b Fn(A)22 b Fl(\012)h Fr(\()p Fn(B)k Fl(\012)c Fr(\()p Fn(C)29 b Fl(\012)22 b Fn(D)s Fr(\)\))p 879 4586 1843 4 v 2639 4584 a Fi(-)1591 4548 y Fm(\013)p Ff(\()p Fm(A;B)s(;C)5 b Fj(\012)p Fm(D)r Ff(\))940 4846 y Fr(\()p Fn(A)23 b Fl(\012)f Fc(K)c Fr(\))28 b Fl(\012)22 b Fn(B)626 b(A)23 b Fl(\012)f Fr(\()p Fc(K)46 b Fl(\012)22 b Fn(B)5 b Fr(\))p 1518 4822 564 4 v 1999 4820 a Fi(-)1650 4785 y Fm(\013)p Ff(\()p Fm(A;)p Fb(K)g Fm(;B)s Ff(\))1663 5241 y Fn(A)22 b Fl(\012)h Fn(B)1212 5032 y Fm(\032)p Ff(\()p Fm(A)p Ff(\))p Fj(\012)p Ff(1)1317 4944 y Fi(Q)1400 5000 y(Q)1483 5055 y(Q)1566 5110 y(Q)1615 5142 y(Q)-83 b(s)2154 5032 y Ff(1)p Fj(\012)p Fm(\025)p Ff(\()p Fm(B)s Ff(\))2200 4944 y Fi(\021)2117 5000 y(\021)2034 5055 y(\021)1951 5110 y(\021)1902 5142 y(\021)g(+)p eop %%Page: 15 7 15 6 bop 1348 236 a Fp(4.)77 b(TENSOR)32 b(PR)n(ODUCTS)1245 b(15)149 425 y Fr(4.)43 b(W)-8 b(rite)32 b Fn(\034)11 b Fr(\()p Fn(A;)17 b(B)5 b Fr(\))28 b(:)g Fn(A)21 b Fl(\012)g Fn(B)33 b Fl(\000)-60 b(!)28 b Fn(B)e Fl(\012)c Fn(A)32 b Fr(for)f Fn(\034)11 b Fr(\()p Fn(A;)17 b(B)5 b Fr(\))28 b(:)g Fn(a)21 b Fl(\012)h Fn(b)28 b Fl(7!)f Fn(b)22 b Fl(\012)f Fn(a)p Fr(.)44 b(Sho)m(w)32 b(that)g Fn(\034)44 b Fr(is)31 b(a)0 541 y(natural)g(transformation)g(\(b)s(et)m(w)m(een)j (whic)m(h)g(functors?\).)44 b(Sho)m(w)33 b(that)513 726 y(\()p Fn(A)22 b Fl(\012)h Fn(B)5 b Fr(\))22 b Fl(\012)g Fn(C)434 b Fr(\()p Fn(B)27 b Fl(\012)c Fn(A)p Fr(\))f Fl(\012)g Fn(C)p 1090 703 369 4 v 1376 701 a Fi(-)1115 665 y Fm(\034)8 b Ff(\()p Fm(X)q(;B)s Ff(\))p Fj(\012)p Ff(1)2463 726 y Fn(B)27 b Fl(\012)c Fr(\()p Fn(A)f Fl(\012)g Fn(C)7 b Fr(\))p 2065 703 V 2351 701 a Fi(-)2227 682 y Fm(\013)p 785 1120 4 351 v 787 1120 a Fi(?)703 959 y Fm(\013)p 2735 1120 V 2737 1120 a Fi(?)2776 961 y Ff(1)p Fj(\012)p Fm(\034)h Ff(\()p Fm(A;C)d Ff(\))513 1213 y Fn(A)22 b Fl(\012)h Fr(\()p Fn(B)k Fl(\012)22 b Fn(C)7 b Fr(\))427 b(\()p Fn(B)27 b Fl(\012)c Fn(C)7 b Fr(\))22 b Fl(\012)g Fn(A)p 1090 1191 369 4 v 1376 1189 a Fi(-)1108 1153 y Fm(\034)8 b Ff(\()p Fm(A;B)s Fj(\012)p Fm(C)d Ff(\))2463 1213 y Fn(B)27 b Fl(\012)c Fr(\()p Fn(C)29 b Fl(\012)22 b Fn(A)p Fr(\))p 2065 1191 V 2351 1189 a Fi(-)2227 1170 y Fm(\013)0 1356 y Fr(comm)m(utes)32 b(for)h(all)d Fn(A;)17 b(B)5 b(;)17 b(C)34 b Fl(2)28 b Fc(K)18 b Fr(-)5 b Fo(Mo)s(d)34 b Fr(and)e(that)1284 1513 y Fn(\034)11 b Fr(\()p Fn(B)5 b(;)17 b(A)p Fr(\))p Fn(\034)11 b Fr(\()p Fn(A;)17 b(B)5 b Fr(\))29 b(=)e(id)2147 1528 y Fm(A)p Fj(\012)p Fm(B)0 1670 y Fr(for)32 b(all)f Fn(A)p Fr(,)h Fn(B)38 b Fr(in)32 b Fc(K)17 b Fr(-)6 b Fo(Mo)s(d)p Fr(.)149 1787 y(5.)44 b(Find)31 b(an)i(example)f(of)g Fn(M)10 b Fr(,)33 b Fn(N)39 b Fl(2)28 b Fc(K)17 b Fr(-)6 b Fo(Mo)s(d)p Fr(-)p Fc(K)56 b Fr(suc)m(h)34 b(that)e Fn(M)h Fl(\012)2619 1802 y Fb(K)2698 1787 y Fn(N)38 b Fl(6)2814 1759 y(\030)2815 1791 y Fr(=)2919 1787 y Fn(N)33 b Fl(\012)3107 1802 y Fb(K)3185 1787 y Fn(M)10 b Fr(.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF