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Fn(\))28 b(=)f Fi(S)6 b Fn(\()p Fi(X)i Fn(\))32 b(for)g(all)f Fi(X)k Fj(2)28 b(C)6 b Fn(.)149 1344 y Fk(Prop)s(osition)36 b(2.2.2.)49 b Fg(L)-5 b(et)46 b Fj(C)52 b Fg(b)-5 b(e)46 b(a)f(c)-5 b(ate)g(gory)46 b(with)g(\014nite)f (\(c)-5 b(ate)g(goric)g(al\))45 b(pr)-5 b(o)g(ducts.)77 b(A)n(n)0 1460 y(obje)-5 b(ct)39 b Fi(G)h Fg(in)g Fj(C)46 b Fg(c)-5 b(arries)39 b(the)h(structur)-5 b(e)41 b Fi(m)c Fn(:)g Fi(G)p Fn(\()p Fg(-)p Fn(\))26 b Fj(\002)g Fi(G)p Fn(\()p Fg(-)p Fn(\))36 b Fj(\000)-58 b(!)38 b Fi(G)p Fn(\()p Fg(-)p Fn(\))h Fg(of)h(a)g(gr)-5 b(oup)39 b(in)h Fj(C)46 b Fg(if)40 b(and)0 1576 y(only)35 b(if)g(ther)-5 b(e)35 b(ar)-5 b(e)35 b(morphisms)f Fi(m)28 b Fn(:)h Fi(G)22 b Fj(\002)h Fi(G)28 b Fj(\000)-58 b(!)29 b Fi(G)p Fg(,)35 b Fi(u)28 b Fn(:)g Fi(E)35 b Fj(\000)-58 b(!)28 b Fi(G)p Fg(,)35 b(and)g Fi(S)f Fn(:)28 b Fi(G)h Fj(\000)-58 b(!)28 b Fi(G)36 b Fg(such)f(that)0 1692 y(the)g(diagr)-5 b(ams)564 1941 y Fi(G)23 b Fj(\002)f Fi(G)g Fj(\002)h Fi(G)210 b(G)22 b Fj(\002)h Fi(G)p 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Fn(\(-)o(\))37 b(:)g Fi(X)8 b Fn(\(-)o(\))37 b Fj(\000)-60 b(!)38 b Fi(Y)21 b Fn(\(-)o(\))38 b(b)m(y)h Fi(f)11 b Fn(\()p Fi(Z)c Fn(\))37 b(=)0 4119 y(Mor)176 4134 y Fh(C)221 4119 y Fn(\()p Fi(Z)r(;)17 b(f)11 b Fn(\).)42 b(In)33 b(particular)e(w)m(e)j(ha)m(v)m(e)g Fi(m)1564 4134 y Fe(X)1659 4119 y Fn(=)28 b(Mor)1939 4134 y Fh(C)1984 4119 y Fn(\()p Fi(X)r(;)17 b(m)p Fn(\))28 b(=)f Fi(m)p Fn(\()p Fi(X)8 b Fn(\).)44 b(The)33 b(diagram)1039 4382 y Fi(G)p Fn(\(-)o(\))22 b Fj(\002)h Fi(G)p Fn(\(-)o(\))f Fj(\002)h Fi(G)p Fn(\(-)o(\))232 b Fi(G)p Fn(\(-)o(\))22 b Fj(\002)h Fi(G)p Fn(\(-)o(\))p 1867 4358 174 4 v 1958 4356 a Ff(-)1859 4317 y Fe(m)1921 4337 y Fn(-)1958 4317 y Fh(\002)p Fd(1)1192 4869 y Fi(G)p Fn(\(-)o(\))g Fj(\002)f Fi(G)p Fn(\(-\))538 b Fi(G)p Fn(\(-)o(\))p 1713 4846 481 4 v 2111 4844 a Ff(-)1904 4805 y Fe(m)1966 4826 y Fn(-)p 1437 4776 4 351 v 1438 4776 a Ff(?)1210 4612 y Fd(1)p Fh(\002)p Fe(m)1362 4632 y Fn(-)p 2314 4776 V 2316 4776 a Ff(?)2355 4605 y Fe(m)2417 4625 y Fn(-)0 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Fn(\))33 b(if)e(and)i(only)f(if)f(the)i(diagram)1320 613 y Fi(G)22 b Fj(\002)h Fi(G)f Fj(\002)g Fi(G)210 b(G)22 b Fj(\002)h Fi(G)p 1823 585 153 4 v 1893 583 a Ff(-)1823 557 y Fe(m)p Fh(\002)p Fd(1)1419 1100 y Fi(G)f Fj(\002)h Fi(G)409 b(G)p 1724 1073 351 4 v 1992 1071 a Ff(-)1868 1052 y Fe(m)p 1556 1002 4 351 v 1557 1002 a Ff(?)1366 845 y Fd(1)p Fh(\002)p Fe(m)p 2141 1002 V 2142 1002 a Ff(?)2181 841 y Fe(m)0 1248 y Fn(comm)m(utes.)43 b(In)33 b(a)g(similar)c(w)m(a)m(y)34 b(one)f(sho)m(ws)h(the)f(equiv)-5 b(alence)33 b(of)f(the)h(other)f(diagram\(s\).)p 3529 1248 4 66 v 3533 1185 59 4 v 3533 1248 V 3591 1248 4 66 v 149 1431 a Fk(Problem)k(2.2.2.)49 b Fn(Let)26 b Fj(C)32 b Fn(b)s(e)26 b(a)f(category)h(with)f(\014nite)g(pro)s(ducts.) 42 b(Sho)m(w)26 b(that)g(a)f(morphism)0 1547 y Fi(f)38 b Fn(:)28 b Fi(G)g Fj(\000)-60 b(!)28 b Fi(G)468 1511 y Fh(0)523 1547 y Fn(in)k Fj(C)39 b Fn(is)32 b(a)g(homomorphism)e(of)i (groups)h(if)e(and)i(only)f(if)1466 1708 y Fi(G)22 b Fj(\002)h Fi(G)311 b(G)p 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Fi(X)8 b Fn(\))27 b(=)h(Mor)1498 2769 y Fh(C)1539 2750 y Fa(op)1610 2754 y Fn(\()p Fi(X)r(;)17 b(G)p Fn(\))27 b(=)h(Mor)2197 2769 y Fh(C)2242 2754 y Fn(\()p Fi(G;)17 b(X)8 b Fn(\),)23 b(suc)m(h)h(that)e(\()p Fi(G)p Fn(\()p Fi(X)8 b Fn(\))p Fi(;)17 b(m)p Fn(\()p Fi(X)8 b Fn(\)\))0 2870 y(is)32 b(a)g(group)h(\(monoid\))d(for)j(eac)m (h)g Fi(X)i Fj(2)29 b(C)6 b Fn(.)149 3054 y Fk(Remark)37 b(2.2.4.)50 b Fn(Let)c Fj(C)52 b Fn(b)s(e)46 b(a)f(category)i(with)e (\014nite)h(\(categorical\))d(copro)s(ducts.)85 b(An)0 3170 y(ob)5 b(ject)35 b Fi(G)f Fn(in)f Fj(C)40 b Fn(carries)34 b(the)h(structure)g Fi(m)c Fn(:)f Fi(G)p Fn(\(-)o(\))23 b Fj(\002)h Fi(G)p Fn(\(-)o(\))30 b Fj(\000)-60 b(!)31 b Fi(G)p Fn(\(-)o(\))j(of)g(a)f(cogroup)h(in)g Fj(C)40 b Fn(if)33 b(and)0 3286 y(only)h(if)f(there)i(are)g(morphisms)e(\001)e (:)g Fi(G)g Fj(\000)-60 b(!)31 b Fi(G)23 b Fj(q)h Fi(G)p Fn(,)35 b Fi(")30 b Fn(:)h Fi(G)g Fj(\000)-60 b(!)31 b Fi(I)8 b Fn(,)35 b(and)f Fi(S)j Fn(:)31 b Fi(G)g Fj(\000)-60 b(!)31 b Fi(G)j Fn(suc)m(h)i(that)0 3402 y(the)d(diagrams)458 4072 y Fi(G)22 b Fj(q)f Fi(G)215 b(G)22 b Fj(q)f Fi(G)i Fj(q)e Fi(G)p 759 4040 157 4 v 833 4038 a Ff(-)763 4019 y Fd(\001)p Fh(q)p Fd(1)556 3585 y Fi(G)410 b(G)22 b Fj(q)f Fi(G)p 661 3552 353 4 v 931 3550 a Ff(-)808 3532 y Fd(\001)p 1177 3970 4 351 v 1179 3970 a Ff(?)1218 3817 y Fd(1)p Fh(q)p Fd(\001)p 592 3970 V 594 3970 a Ff(?)496 3817 y Fd(\001)1822 4070 y Fi(G)i Fj(q)e Fi(G)177 b(I)30 b Fj(q)22 b Fi(G)2546 4043 y Fj(\030)2547 4074 y Fn(=)2652 4070 y Fi(G)2756 4043 y Fj(\030)2757 4074 y Fn(=)2861 4070 y Fi(G)h Fj(q)e Fi(I)p 2124 4040 120 4 v 2161 4038 a Ff(-)2123 4019 y Fe(")p Fh(q)p Fd(1)1920 3585 y Fi(G)849 b(G)22 b Fj(q)g Fi(G)p 2026 3552 792 4 v 2735 3550 a Ff(-)2393 3532 y Fd(\001)p 2981 3970 4 351 v 2983 3970 a Ff(?)3022 3817 y Fd(1)p Fh(q)p Fe(")p 1957 3970 V 1959 3970 a Ff(?)1861 3817 y Fd(\001)2386 3816 y(1)2061 3674 y Ff(Q)2144 3730 y(Q)2227 3785 y(Q)2310 3840 y(Q)2393 3896 y(Q)2476 3951 y(Q)2505 3970 y(Q)-83 b(s)1372 4236 y Fi(G)326 b(I)p 1477 4203 269 4 v 1663 4201 a Ff(-)2152 4236 y Fi(G)p 1855 4203 V 2041 4201 a Ff(-)1972 4183 y Fe(")1274 4723 y Fi(G)22 b Fj(q)g Fi(G)507 b(G)22 b Fj(q)f Fi(G)p 1575 4691 451 4 v 1943 4689 a Ff(-)1733 4670 y Fd(1)p Fh(q)p Fe(S)1733 4754 y(S)t Fh(q)p Fd(1)p 1408 4621 4 351 v 1410 4621 a Ff(?)1312 4468 y Fd(\001)p 2188 4621 V 2190 4353 a Ff(6)2229 4468 y Fh(r)0 4867 y Fn(comm)m(ute)28 b(where)i Fj(r)e Fn(is)g(dual)g(to)g(the)h(morphism) e(\001)i(de\014ned)h(in)e(A.2.)42 b(The)30 b(m)m(ultiplications)24 b(are)0 4983 y(related)32 b(b)m(y)i(\001)542 4998 y Fe(X)637 4983 y Fn(=)28 b(Mor)917 4998 y Fh(C)962 4983 y Fn(\(\001)p Fi(;)17 b(X)8 b Fn(\))27 b(=)h(\001\()p Fi(X)8 b Fn(\).)149 5099 y(Let)40 b Fj(C)46 b Fn(b)s(e)39 b(a)g(category)h(with)f(\014nite) g(copro)s(ducts)h(and)g(let)e Fi(G)i Fn(and)f Fi(G)2778 5063 y Fh(0)2841 5099 y Fn(b)s(e)g(cogroups)h(in)f Fj(C)6 b Fn(.)0 5216 y(Then)37 b(a)e(homomorphism)e(of)i(groups)h Fi(f)43 b Fn(:)33 b Fi(G)1694 5179 y Fh(0)1750 5216 y Fj(\000)-60 b(!)32 b Fi(G)k Fn(is)f(a)g(morphism)f Fi(f)43 b Fn(:)33 b Fi(G)g Fj(\000)-60 b(!)32 b Fi(G)3143 5179 y Fh(0)3202 5216 y Fn(in)j Fj(C)41 b Fn(suc)m(h)p eop %%Page: 43 5 43 4 bop 931 236 a Fl(2.)76 b(MONOIDS)32 b(AND)g(GR)n(OUPS)g(IN)h(A)g (CA)-6 b(TEGOR)g(Y)827 b(43)0 425 y Fn(that)32 b(the)h(diagram)1470 547 y Fi(G)311 b(G)22 b Fj(\002)h Fi(G)p 1576 519 254 4 v 1747 517 a Ff(-)1673 498 y Fd(\001)1458 1037 y Fi(G)1535 1001 y Fh(0)1835 1037 y Fi(G)1912 1001 y Fh(0)1957 1037 y Fj(\002)g Fi(G)2134 1001 y Fh(0)p 1587 1007 219 4 v 1723 1005 a Ff(-)1656 985 y Fd(\001)1715 962 y Fc(0)p 1994 936 4 351 v 1996 936 a Ff(?)2035 778 y Fe(f)7 b Fh(\002)p Fe(f)p 1507 936 V 1508 936 a Ff(?)1428 778 y Fe(f)0 1142 y Fn(comm)m(utes.)43 b(An)33 b(analogous)f(result)g (holds)g(for)g(comonoids.)149 1317 y Fk(Remark)37 b(2.2.5.)50 b Fn(Ob)m(viously)33 b(similar)e(observ)-5 b(ations)34 b(and)g(statemen)m(ts)h(can)f(b)s(e)g(made)f(for)0 1433 y(other)38 b(algebraic)f(structures)j(in)d(a)h(category)h Fj(C)6 b Fn(.)60 b(So)39 b(one)f(can)h(in)m(tro)s(duce)f(v)m(ector)h (spaces)h(and)0 1549 y(co)m(v)m(ector)46 b(spaces,)i(monoids)43 b(and)h(comonoids,)i(rings)e(and)g(corings)g(in)f(a)h(category)h Fj(C)6 b Fn(.)78 b(The)0 1666 y(structures)44 b(can)e(b)s(e)g(describ)s (ed)h(b)m(y)g(morphisms)e(in)g Fj(C)48 b Fn(if)41 b Fj(C)48 b Fn(is)42 b(a)g(category)g(with)g(\014nite)f(\(co-\))0 1782 y(pro)s(ducts.)149 1956 y Fk(Problem)36 b(2.2.3.)49 b Fn(Determine)37 b(the)h(structure)i(of)d(a)g(co)m(v)m(ector)j(space)f (on)e(a)h(v)m(ector)h(space)0 2072 y Fi(V)54 b Fn(from)31 b(the)i(fact)g(that)f(Hom)o(\()p Fi(V)5 b(;)17 b(W)d Fn(\))32 b(is)g(a)h(v)m(ector)g(space)h(for)e(all)f(v)m(ector)i(spaces) h Fi(W)14 b Fn(.)149 2247 y Fk(Prop)s(osition)36 b(2.2.6.)49 b Fg(L)-5 b(et)39 b Fi(G)34 b Fj(2)h(C)45 b Fg(b)-5 b(e)38 b(a)g(gr)-5 b(oup)38 b(with)g(multiplic)-5 b(ation)38 b Fi(a)25 b Fj(\003)g Fi(b)p Fg(,)39 b(unit)g Fi(e)p Fg(,)h(and)0 2363 y(inverse)k Fi(a)396 2327 y Fh(\000)p Fd(1)536 2363 y Fg(in)h Fi(G)p Fn(\()p Fi(X)8 b Fn(\))p Fg(.)76 b(Then)45 b(the)g(morphisms)f Fi(m)j Fn(:)h Fi(G)29 b Fj(\002)i Fi(G)47 b Fj(\000)-58 b(!)47 b Fi(G)p Fg(,)h Fi(u)f Fn(:)g Fi(E)53 b Fj(\000)-58 b(!)48 b Fi(G)p Fg(,)g(and)0 2479 y Fi(S)33 b Fn(:)28 b Fi(G)g Fj(\000)-58 b(!)28 b Fi(G)35 b Fg(ar)-5 b(e)35 b(given)f(by)968 2636 y Fi(m)28 b Fn(=)f Fi(p)1233 2651 y Fd(1)1295 2636 y Fj(\003)22 b Fi(p)1416 2651 y Fd(2)1455 2636 y Fi(;)216 b(u)27 b Fn(=)h Fi(e)1930 2651 y Fe(E)1990 2636 y Fi(;)216 b(S)33 b Fn(=)28 b(id)2511 2593 y Fh(\000)p Fd(1)2511 2661 y Fe(G)2605 2636 y Fi(:)149 2810 y Fb(Pr)n(oof.)42 b Fn(By)35 b(the)f(Y)-8 b(oneda)34 b(Lemma)e(A.9.1)i(these)h(morphisms)e(can)h(b)s (e)g(constructed)h(from)0 2927 y(the)28 b(natural)f(transformation)f (as)i(follo)m(ws.)40 b(Under)29 b(Mor)2098 2942 y Fh(C)2143 2927 y Fn(\()p Fi(G)12 b Fj(\002)g Fi(G;)17 b(G)12 b Fj(\002)g Fi(G)p Fn(\))30 b(=)d Fi(G)12 b Fj(\002)g Fi(G)p Fn(\()p Fi(G)g Fj(\002)g Fi(G)p Fn(\))3523 2899 y Fj(\030)3523 2931 y Fn(=)0 3057 y Fi(G)p Fn(\()p Fi(G)j Fj(\002)g Fi(G)p Fn(\))g Fj(\002)g Fi(G)p Fn(\()p Fi(G)g Fj(\002)g Fi(G)p Fn(\))1024 3001 y Fh(\003)961 3057 y Fj(\000)-16 b(!)27 b Fi(G)p Fn(\()p Fi(G)15 b Fj(\002)g Fi(G)p Fn(\))27 b(=)h(Mor)1870 3072 y Fh(C)1915 3057 y Fn(\()p Fi(G)15 b Fj(\002)g Fi(G;)i(G)p Fn(\))28 b(the)h(iden)m(tit)m(y)g(id)3003 3072 y Fe(G)p Fh(\002)p Fe(G)3200 3057 y Fn(=)f(\()p Fi(p)3391 3072 y Fd(1)3430 3057 y Fi(;)17 b(p)3523 3072 y Fd(2)3562 3057 y Fn(\))0 3173 y(is)44 b(mapp)s(ed)h(to)f Fi(m)49 b Fn(=)f Fi(p)932 3188 y Fd(1)1002 3173 y Fj(\003)30 b Fi(p)1131 3188 y Fd(2)1171 3173 y Fn(.)80 b(Under)45 b(Mor)1761 3188 y Fh(C)1806 3173 y Fn(\()p Fi(E)6 b(;)17 b(E)6 b Fn(\))48 b(=)h Fi(E)6 b Fn(\()p Fi(E)g Fn(\))48 b Fj(\000)-60 b(!)49 b Fi(G)p Fn(\()p Fi(E)6 b Fn(\))48 b(=)g(Mor)3280 3188 y Fh(C)3325 3173 y Fn(\()p Fi(E)6 b(;)17 b(G)p Fn(\))0 3290 y(the)40 b(iden)m(tit)m(y)g(of)g Fi(E)46 b Fn(is)39 b(mapp)s(ed)h(to)f(the)i(neutral)e(elemen)m(t)h Fi(u)g Fn(=)g Fi(e)2534 3305 y Fe(E)2594 3290 y Fn(.)65 b(Under)41 b(Mor)3165 3305 y Fh(C)3210 3290 y Fn(\()p Fi(G;)17 b(G)p Fn(\))40 b(=)0 3409 y Fi(G)p Fn(\()p Fi(G)p Fn(\))27 b Fj(\000)-60 b(!)28 b Fi(G)p Fn(\()p Fi(G)p Fn(\))g(=)f(Mor)939 3424 y Fh(C)984 3409 y Fn(\()p Fi(G;)17 b(G)p Fn(\))32 b(the)h(iden)m(tit)m(y)g(is)f(mapp)s(ed)g(to)g(its)g Fj(\003)p Fn(-in)m(v)m(erse)i Fi(S)f Fn(=)28 b(id)3227 3366 y Fh(\000)p Fd(1)3227 3434 y Fe(G)3321 3409 y Fn(.)p 3529 3409 4 66 v 3533 3346 59 4 v 3533 3409 V 3591 3409 4 66 v 149 3583 a Fk(Corollary)36 b(2.2.7.)50 b Fg(L)-5 b(et)40 b Fi(G)d Fj(2)g(C)46 b Fg(b)-5 b(e)40 b(a)f(c)-5 b(o)g(gr)g(oup)40 b(with)g(multiplic)-5 b(ation)39 b Fi(a)26 b Fj(\003)g Fi(b)p Fg(,)41 b(unit)f Fi(e)p Fg(,)i(and)0 3699 y(inverse)k Fi(a)398 3663 y Fh(\000)p Fd(1)540 3699 y Fg(in)g Fi(G)p Fn(\()p Fi(X)8 b Fn(\))p Fg(.)81 b(Then)46 b(the)h(morphisms)e Fn(\001)50 b(:)h Fi(G)e Fj(\000)-58 b(!)51 b Fi(G)31 b Fj(q)g Fi(G)p Fg(,)50 b Fi(")f Fn(:)h Fi(G)g Fj(\000)-58 b(!)51 b Fi(I)8 b Fg(,)50 b(and)0 3815 y Fi(S)33 b Fn(:)28 b Fi(G)g Fj(\000)-58 b(!)28 b Fi(G)35 b Fg(ar)-5 b(e)35 b(given)f(by)999 3972 y Fn(\001)28 b(=)f Fi(\023)1245 3987 y Fd(1)1308 3972 y Fj(\003)21 b Fi(\023)1413 3987 y Fd(2)1453 3972 y Fi(;)216 b(")28 b Fn(=)f Fi(e)1918 3987 y Fe(I)1958 3972 y Fi(;)216 b(S)34 b Fn(=)27 b(id)2480 3929 y Fh(\000)p Fd(1)2480 3998 y Fe(G)2574 3972 y Fi(:)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF