Rosický, Jiří

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeJiří RosickýOne obstruction for closednessCommentationes Mathematicae Universitatis Carolinae, Vol. 18 (1977), No. 2, 311--318Persistent URL: http://dml.cz/dmlcz/105775Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czOOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 18,2 (1977) ONE OBSTRUCTION K)R CLOSEDNESS Jiří BOSICKÍ, Brno Abstract: There is stated one necessary condition for a fory to gebras. categ be closed. Applications are given to categories al  Key words: Closed category, adjoint on the right, vari­ety of algebras, semigroup. AMS: Primary 18D15 Ref. 2.: 2.726.11,2.725.2 Secondary 08A15» This note deals with symmetric monoidal closed categories in the sense of £1]. In what follows, we will call them brief-ly closed categories^ It i9 well-known (see £31) that a vari-ety V of universal algebras carries the structure of a clos-ed category with the free algebra on one generator as the unit if and only if it is commutative. It means that for each n-ary operation f and for each algebra V the map f: An—.• A is a ho-momorphism, i.e. for any m-ary operation g: A m — • A it holds? gtr -* fg • However, any variety of unary algebras is cartesian closed and thus there are non-commutative closed varieties^ Recently, Foltz and Lair have deduced in 1.2.3 one necessary condition for olosedness and they have shown that the variety of groups is not closed* We will give another obstruction for clo9edne99 which 9hows (in the same way as the criterion of - 311 [ Zl) that the var ie t i ea of grupoid8 with unit and rings are not c losed . The author i s indebted to Ch. Lair and A. Fultr for a valuable conversation . 1 . The obatruction theorem. Let V be a category. In what fo l lows , we w i l l suppose that 1 i s an object of V such that every object of V i s an i terated colimit of copies of E. I t implies that the functor / / » V ( E , - ) : V — > Set i s f a i t h f u l . Def ini t ion 1: A couple ( F , y ) w i l l be cal led a connec-t i o n on a category V i f F: V o p — > V ia a functor and <f a y v w : V (V,FW) —> V (WfiV) a natural isomorphism such that cf^ T . cpv w =- 1 for any VfW e V -A morphism of connections oc : ( F , < p ) — ^ ( G , ^ ) i s a na-tural transformation eC : F—>G such that ^ f* V (%9oo^) -» -* V Cwfctfv). 9 V w for any VfW c V * In thia way we get the category C(V ) of connections on V • I f (F,cp) i s a connection on V , then F la adjoint on the right with i t a e l f , where cp ia the adjunction isomorph-ism. Our terminology i s an adaptation for our purposes! of the terminology of I a b e l l [ 4 1 , where a connection from a catego-ry A to a category B ia a contravariant functor F from A to B having an adjoint on the r ight G* Under certa in 8uppos i t i -ons such connections can be i d e n t i f i e d with A-objecta i n B# Choosing objects a e A and b e B one geta underlying objec t s F a t B and Gbe A. So connections i n our sense can be i d e n t i -f i e d with double V - o b j e c t a ( i . e . with V -objecta in V ) auch that both underlying objects are iaomorphic. Later, - 312 such an underlying object wil l be called pseudocommutative. Denote by V the category having objects (V ft) f where t : /V/—> /V/ i s a mapping such that t - 1 and morphisms f: (Vft) —+ (V' ft ' ) where f: V — * v ' i s a morphism in V and t 7 f / =* / f / t . Define a functor U: C(V ) —> V by U(F,<y ) a (FEf 9 E E) and Uoo =- <*-£. The following proposition i s , in fact, a corollary of Theo-rem 3.8 from C43. Proposition 1: U i s fu l l and faithful . Proof: U i s faithful following the property of E and the fact that connections take colimits to l imits. I f f: U(Ff<3p)~* —»U(Gf<3f) i s a morphism in V f then -A fa: V lEfFX) - - * V (X,FE) ' >V (X,GE) ' > V(EfGX) are components of a natural transformation (Z : /F/— .•/GA Following the proof of Theorem 3.8 of [43 f can be extended by colimits to a natural transformation 00 : F—> G such that GCj =* f and /<x/ * ft . It immediately implies that T l W* ^ fBjQC Î - V(W, oc-g), cf ^ w . Since V (irfG-) takes colimits and every object of V is an iterated colimit of copies of Ef oc have to be a morphism of connections. Definition 2: An object V of V will be called pseudo-commutative (with respect to E) i f there i s a connection ( F , y ) on V such that V -* Fl# So, a pseudocommutative object V i s , under certain suppo-sitions on V , and underlying object of a double V -object such that the second underlying object i s isomorphic to V v i s 2 isomorphism t such that t -» 1. Clearly, an object isomorphic - 313 -to a pseudocommutative one i s paeudocommutative. Proposition 2: Let V have products. Then a product of pseudocommutative objects is pseudocommutative. Proof: Let (F i f Cf^) be connections on V . Clearly F ~ (. TT F i f #TTT Cf$) i s their product, which implies the assertion. Let V be a closed category. Denote by V ( - , - ) : 2 V° P x V—* V the internal horn-functor, by - ® - : V x V • V the tensor product, by p » Py w x : - V(V®WfX) —> V (V, V(WfX)) the adjunction isomorphism, by c «Cy w: ? ® l - ^ I 0 ? the symmetry and by I c V the unit. Then ( ^ (- fX) f Pw v ^ V (cfX).p^ w x ) i s a connection on V for any X * V * Theorem 1: Let V be a closed category with products. Then any object of V ia isomorphic to a subobject of a pse-udocommutative object. Proof: Since I i s an iterated colimit of copies of Ef V (I,V) is an iterated limit of copies of V(EfV) for any V 6 V • Using the construction of limits by products and equalizers and Proposition Z we get that V (IfV) i s a sub-* object of a pseudocommutative object* But V is isomorphic to V ( l f V ) . Remark: 1) The same assertion holds more generally for symmetric (non-monoidal) closed categories. These are (non-monoidal) closed categories endowed with a symmetry s * s SV t Z: V (V» ^ ( W , Z ) ) — * V ( l M ( V f Z ) ) satisfying ap-propriate axioms (see 151). - 314 2) If V is an (epi-extremally mono)-category, then the word aubobjeot can be replaced in Theorem 1 by an extre-mal subobject. Another obstruction for closedness is stated in £2J» Roughly speaking, it asserts that if V is closed and any double V -object underlies a triple one, then any object of V underlies a double one. This result can be strengthened in the sense that if any pseudocommutative object underlies a triple one, then any object underlies a double one. 2. Closed varieties of algebras. Let V be a variety of universal algebras and let S be the free algebra on one generator. Then / / is the usual forgetful functor. Proposition 3: An algebra V £ V is pseudocommutative if and only if there is a bijection t: / V / — > / ! / such that t2* =*1 and for any n-ary algebraic operation h: /V/ n— . * / V / the mapping thtn: /V/ n—*VV/ is a homomorphism. Clearly, tht are algebraic operations of a new algebra on /V/, which is isomorphic to V via t» Denote by Ps(V) the full subcategory of V consisting of objects (V,t) such that t makes V to be pseudocommutative* Then Ps(V) is a new variety which arises from V by adding a new unary operation t and the axioms given by Proposition 3. Hence the forgetful functor H: Ps(V ) — > V has a left adjoint L. Corollary 1: Let the variety V admit a structure of a closed category. Then the unit t£ of the adjunction L —I H is a monomorphism. - 315 -Proof: Since an object of V isomorphic to a objects from Ps(V J belongs to Ps(\Y ) , by Theorem 1 there i s a mono-f morphism V — > HW. Hence ^ ^ i s mono because i t f ac tor i ze s through f. Since Ps(V ) i s a aubvariety of V , L i s a sui table quotient of the l e f t adjoint to the forget ful functor V —* —> V . This l e f t adjoint assigns t o each V6 V and ob-»-v j ec t (w,t) 6 V where the underlying set /W/ of W i s the union of a chain XQs X-,5 X^SX-jS . . . . Here XQ = / V / t X^ 1m the coproduct / V / U / V / , X^ i s the underlying set of an algeb-ra in V free over X^but with the algebraic structure of V on X0, X-j = XgU (Xg - X^) and so on. The algebraic s truc-ture of W i s c lear and t i s given by: t/X^ interchanges the both copies of / V / , t / X j - X̂  interchanges the both copies of Xp - X^ and so on. This procedure i s caused by the fact that V i s the pullback in CAT of V -—> S«t and of the forget fu l functor of the category of algebra* 2 with the one unary operation t such that t « 1* L y i e l d s one general construction of pseudocommutative algebras. Another one i s given by the following lemma* Lemma 1: Let V/j_ and V^ be two algebras of V on the same underlying se t X such that operations of V^ are homomor-phisms of Vg, ( i . e . (%,V2) i s a double V - o b j e c t ) . Then t : :XxX— .»XxX, t(x^,X2s) * ( x ^ x ^ ) , makes V ^ ? 2 to be pseudo-commutative. Proof i s straightforward . Example 1 (see C21): Let V be the variety of groups. 316 It i s well-known that double groups are commutative. Thus V i s not closed. The same argument applies to the variety of semigroups (or grupoids reap.) with the unit. Example 2: Let V be the variety of ringe. It ia ea-sy to show that any double ring has the zero multiplication. Thus V i s not closed. Example 3: Let V be the variety of semigroups. A se -migroup V ia pseudocommutative i f f there i s a bijection t : : /V/~~* A / such that X2 » 1 and t ( t (xy) t (x 'y ' ) ) • • t ( t ( x ) t ( x ' ) ) t ( t ( y ) t ( y ' ) ) for any x,y,x',y'eV» Since the left semigroup on X forms a double semigroup with any semi-group on X, by .Lemma 1 /Vx V/ carries the following pseudo-commutative semigroup V* for any semigroup V: (u/pUgHv^Vg)-* = (u^v^fUg). Evidently V i s a subsemigroup of V* and thus we do not need Corollary 1 for the verification that V overco-mes Theorem 1 . One i s tempted to try what gives the assign-ment V (E,V) 9 v* . This functor V (E,-): V —* V has a left adjoint - ® E given as follows. Let «* be the transi-tive hull of the relation ~ on /V/ such that u^t i f f the-re are WfŴ fWge V such that u * wŵ f v ~ ww2» Then V® E i s the coproduct of V and of a free semigroup on /V/ /** . Sin-ce V® B a E for no semigroup V, we do not get a closed stru-cture on V . The author does not know whether such a struc-ture can exist . R e f e r e n c e s Cl] S. EILENBERG and G.M. KELLT: Closed categories, Eroe. Int. Conf. Categ. Alg. (La Jolla 1965),Springer-317 Verlag 1966. 123 F. FOLTZ and Ch. LAIR: Fermeture standard des catego­ries algébriques, Cahiers Topo. Géo. Diff. XIII-3 (1972), 275-307. [33 P» FREYD: Algebra valued functors in general and tensor products in particular, Colloq. Math* 14 (1966), 89-106. [43 J«R# ISBELL: General functorial semantics, Amer. J. Math. 94(1972), 535-596. [53 W.J. de SCHIPPER: Symmetric closed categories, Math. Centre Tracts 64, Amsterdam 1975. Přírodovědecká fakulta UJEP Janáčkovo nám. 2a 662 95 Brno Československo (Oblátům 7*3. 1977) 318 -

One obstruction for closedness

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One obstruction for closedness

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Institute of Mathematics AS CR, v. v. i.