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C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M ANFRED B ERND W ISCHNEWSKY Aspects of categorical algebra in initialstructure categories Cahiers de topologie et géométrie différentielle catégoriques, tome 15, no 4 (1974), p. 419-444 <http://www.numdam.org/item?id=CTGDC_1974__15_4_419_0> © Andrée C. Ehresmann et les auteurs, 1974, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Vol. XV-4 CAHI ERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE ASPECTS OF CATEGORICAL ALGEBRA IN INITIALSTRUCTURE by Manfred an Bernd WISCHNEWSKY F: K - L, the Initialstruc ture functors of BOURBAKI’s notion of CATEGORIES* « initial categorical generalization objects [3], equivalent to Kenni- pullback stripping functors, which Wyler calls Top-functors, reflect almost all categorical properties from the base category L to the initial structure category K briefly called INS-category [1,4,5,6,8,11,12,13, 13,16,18,19,20,21,22,37]. So for instance if L is complete, cocomplete, wellpowered, cowellpowered, if L has generators, cogenerators, proj ectives, injectives, or ( coequalizer, mono )-bicategory structures ( ==> homomorphism theorem ) , then the same is valid for any INS-category over L . Finally all important theorems of equationally defined universal algebra ( e.g. existence of free K-algebras, adjointness of algebraic functors... ) can be proved for algebras in INS-categories, if they hold for algebras in L . The most well known INS-categories over Ens are the categories of topological, measurable, limit, locally path-connected, uniform, compactly son’s generated, or zero dimensional spaces. This survey article deals with the braic - - three types of over of PFENDER algebraic categories sense of alge- with INS-categories, namely algebraic categories of Z-continuous functors [29,30,33] , monoidal algebraic categories over monoidal base categories categories sense - following in the [35], in the sense of T H I E B A U D [36] resp. in the EILENBERG-MOORE. These three types of «algebraic categories» include comma-catego- categories of finite algebras, locally presentable categories, ElLENBERG-MOORE: categories, categories of monoids in monoidal categories. ries, algebraic categories in the * Conference donn6e au sense of LAWVERE, Colloque d’Amiens ( 1973) 419 In this paper it is shown that each of the above ries is algebraic catego- Hence the whole theo- INS-category again INS-category. here the first part, can again be applied in INS-categories, presented to algebraic categories over INS-categories. In particular it is shown that this implies that adjointness of « algebraic» functors over L induces adjointness of « algebraic » functors over an INS-category K . Since furthermore together with K also K°P is an INS-category, and since the algebras in K°P are just the coalgebras in K , the theory is also valid for coalgebras in INS-categories. Moreover I will show that all the basic results on algebraic categories even hold for reflective or coreflective subcategories of INS-categories if they hold for the corresponding INS-categories. So over an an ry of for instance let U be a coreflective or even an epireflective of a coreflec- subcategory of the category of topological spaces and continuous mappings like the categories of compactly generated, locally path connec- tive ted, 7-spaces, or i = 0, 1, 2, 3, and let Z C [C ,S ] be a set of functo- morphisms in [C ,S]. Then the algebraic categories Z ( C , U ) Z -continuous functors with values in U are complete, cocomplete, rial powered, and cowellpowered. adjoint. The gebras in U . are 1. same of all well- Furthermore all inclusion functors holds for the category Z (C, U°p ) for all Z-coal- INS-functors, INS-categories, and INS-morphisms. Let F: K - L be a functot. Let I be an arbitrary necessarily resp. by AL : not discrete, small category, and denote by Ak : K-> [I,K] L -> [I, L ] the corresponding diagonal functors, and for T E [I, K ] the comma categories If there is no (AK, T) resp. confusion, I’ll write of all functorial for A K resp. A L simply A. weakly isomorphisms. With transportabl e if F give the following creates 1.1. Let F:K - L be D E F I N I T IO N . (AL , FT) morphisms F is called this notation we can faithful, fibre-small functor. F i s called 420 INS-functor, an and K and T 1, and for all functors has an is over L , if for all small categories the functor with counit cone, then the cone a GT qj : Al* -> T is called an INS - INS-object generated by qj - If t,: I - F k is a monoGO: l* -> k is called an embedding, and l* an INS-subob- and l* E K an morphism, then ject of k . Dually one 1.2. E [1, K J adjoint ( = right adjoint) GT If Y : Al -> F T cone, INS-category EXAMPLES: category of sets defines coinitialstructure The following categories functor,..... with obvious INS-functors: the uniform, measurable, or completely regular, Let us now of topological, principal limit compactly generated spaces, categories based spaces, of Borel spaces, of spaces, of spaces with bounded structure, of of limit, over S the INS-categories are or zero dimensional spaces.... recall G R O T H E N D I E C K’ s construction of a split cate- completely ordered sets with suprema preserving mappings. Then there corresponds by GROTHENDIECK ( [8]) to each functor P : L°p -> Ord (V) a split category Fp : Kp - L : the objects of Kp are the pairs (l, k) with k E P l and /6L. The Kp-morphisms f:(l,k)-> (I’,k’) are exactly those L-morphisms f:1-1’ with P f k k’ . The functor Fp : K p - L is the projection (l,k) l->H . gory. Let Ord (V) be the category of Herewith we can give the following [1,8,11,15,20,23]. Let F: Ksmall functor. Regard the following assertions : ( i ) F is an INS-functor. ( ii ) F°p is an INS-functor. 1.3. THEOREM ( iii ) The category ( K, F) over L is L -L-equivalent be a faithful, fibre- to a split category ( Kp , Fp ). ( iv ) F preserves limits and has a full 421 and faith ful adjoint J . Then (i) ( ii ) ==> (iii) ==>(iv). 1 f plete, then (iv) ==>(i). ==> The ANTOINE equival’ence ( i ) [1] ==> (ii) ROBERTS [15], and L and K moreover ==> ( iii ) are com- proved by straightforward from was but follows first categories [8]. The characterization ( iv ) ==>(i) was first given by HOFFMANN [11]. A proof of ( iv ) ==> (i), in some way simpler than the original one, can immediately be obtained from both the following statements : GROTHENDIECK’s 1) A cone an fibied is INS-cone if and only an cone INS-cone, 2 ) Let K, L be preserving tion on ( l* , Y : Al* -> T), TE [I, K], if the induced is paper categories functor with (cf. [14] with Let F: K - L be an inverse. Then F is pullbacks. adjoint right a pullback a fibra- Corollaire 3.7) (cf. Theorem 1.9a). Hence, if F fulfills (iv), F is by 2 and thus together with 1 an INS-functor provided K and L are complete. The full and faithful right adjoint of an INS-functor is obtained from definition 1.1 by regarding the void category as index category. The objects of the image category of this functor are called codiscre te Kob je cts. 1.4. we [18,20,22]. following assertions : PROPOSITION have the a 1 ) K-limits ( K-colimits ) fibration Let F:K-L be an INS-functor. Then limits ( colimits ) in L supplied with the «initialstructure» ( coinitialstructure ) generated by the projections ( injections ). 2 ) F preserves and reflects monos and epis. 3 ) K is wellpowered (cowellpowered) if and only if L has this proare perty. 4 ) K has generators, cogenerators, projectives, ly if K has such objects. 422 or injectives if and on- 5 ) K is a ( coequalizer, mono )-bicategory if and only if L is such a category. Let K -> K’ 1.5. as now well F:K-L and F’:K’-L’ be INS-functors, and let M : N:L-L’ be as [22,23,37]. DEFINITION called an arbitrary INS-morphism The if both of the functors. pair (NI, N): (K, F)->(K’, F’) is following conditions hold : 1) NF=F’M, 2 ) for all small categories I and all functors 7B’/-’K right inverses G’MT gram commutative up resp. to an GT of ’ F’MT resp. F?.. make the adjoint following diathe isomorphism pair (M, N) preserves INS-cones. In WYLER’s language of Top-categories, i.e. reduced INS-categories the INS-morphisms correspond to his taut liftings [22,23] . In this case we say that the [19] 1.6. REMARKS. 1) The category Initial of all INS-functors and INS-morphisms is ble category in the sense of a dou- EHRESMANN. 2) The category Initial (L) of all INS-categories and INS-morphisms over a constant base category L is canonically isomorphic to the full functor category (L°p Ord (V] ( cf. [19]). 1.7. THEOREM [22 , 23]. Suppose that (M, N) : (K,F) -> ( K’ F’) be an INS-morphism. Then ( i ) Af is adjoint. ( ii ) N is adjoint. we have the following equivalent The statements : implication (i) ==> (ii) is trivial by theorem 1.3 ( iv ) . In case categories involved are complete, it suffices to find a soluset for tion M, since M preserves obviously limits by proposition 1.4 and definition 1.5 . If R : L’ - L denotes a coadjoint of N, then for all that all 423 k’ E K’ the fibre F-1 (RF’(k’)) is a solution for k’, set as one sees immediately by factorizing F’f : F’k’- N F k, f : k’->M k E K’, through the unit of (R -l N), and then by supplying R F’ k’ with the INS structure... In the general case one takes the infimum of all objects in this fibre appearing in any of the above factorizations. For the INS-categories 1.8. rest are DEFINITION. assume now that all base of categories complete. Let F:K-L, resp. F’:K’-L’, be INS-functors. De- J’ , adjoint right inverse functor of F , resp. F’ , and by O : Id -> J F, resp. O’: Id -> J’ F’, the corresponding units. Let furtherN F . The more M : K - K’ and N : L - L’ be a pair of functors with F’ M pair (M, N) preserves codiscrete objects if note by J , of this paper resp. an = pair o f functors betINS-functors. 1 f ( M, N ) preserves codiscrete objects, the following 1.9. THEOREM. Let ( M, N ) : ( K, F ) -> ( K’, F’ ) be ween statements are a equival ent : (i) (M, N): (K, F)->(K’, F’) is an INS-morphism. ( ii ) M and N preserve limits. The cones and implication (i) ==>(ii) is obvious, since hence in particular limit-cones. Thus one (ii) ===>(i) . M preserves INS- has only From the remark 1 of theorem 1.3 it follows that to prove one has only to show that ( M , N ) preserves INS-cones of the form (l*, Y: 1* - k ) generated by F Y : l -> Fk , i.e. ( M , N ) is a morphism of fibration in the sense of GROTHENDIECK. But this follows immediately from the fact, that a) the INS-cone Y : l* -> k where O : Id -> J F is a projection denotes the unit of the 424 of adjoint a pullback, namely functor pair ( F, J ), and ia’* denotes the b) ( M , N ) preserves 1.10. induced by idl : l -> l . codiscrete objects, and K-morphism EXAMPLE [18,20,cp.22,36,37]. in the M preserves Let A be of L A W V E R E and let F : K -> L be sense an pullbacks., algebraic theory INS-functor an over a complete category L . Then we obtain the following commutative diagram of limit preserving functors between complete categories denote the obviously particular : all corresponding forgetful assumptions of theorem 1.9. VK is adj oint VK resp. VL fulfills I.II. COROLLARY. 1.12. EXAMPLE. Each Let A be Then the functor if and morphism an Alg ( A , - ) only if in Initial ( algebraic theory defines in an VL is functors. Hence L ) is adjoint. in the sense obvious way Alg ( A , H ) is an INS-morphism, it is coadjoint by lary. A typical example for this situation is the functor induced by the 1.13. DEFINITION a strictly U is closed 20132013201320132013> topological - topological a of LAWVERE. functor [19] : the above Corol- groups INS-morphism uniform spaces be obtain in adjoint. Since uniform groups we (VK, V L) [19,23]. Let F: K - L be subcategory of K under INS-objects, i.e. full an INS-functor, and let . U is called if for all small 425 spaces. U INS-subcategory if categories I and all an T E [I, U] functors lies the INS-object again in U . INS-subcategories 1.14. [19]. can l* generated by be characterized in the Let F: K - L be (Y : Al -> FT) a cone following way: INS-functor. A strictly full subcategory U of K is an I NS-subcategory of K if and only if 1 ) U is closed under products and INS-subobjects, 2 ) U contains all codiscrete K-objects. THEOREM an EXAMPLES: INS-category over S . Since K is complete, wellcowellpowered, a stricly full subcategory U of K is epi- 1 ) Let K be powered, and an reflective in K if and only if U objects [9,18 19] . Hence we is closed under products and INS-sub- get the COROLLARY [19]. Let K be an INS-category over S, and le t U strictly full subcategory of K containing all codiscrete K-objects. there are equivalent : ( i ) U is an INS-subcategory of K . ( ii ) U is epire f lective i n K . be a Then K-object of an INS-category over S are generators in K , we obtain by dualizing and applying the results of HERRLICH -STRECKER [101 the following 2 ) Since all but one discrete COROLLARY [19,23]. Let K be an INS-category over S, and let a strictly full subcategory of K containing all discrete K-objects. there equivalent : ( i ) U is a COINS- su bcatego ry o f (ii) U is coreflective in K . U be Then are In of case these corollaries Finally K =r x’op one K. can find a whole host of examples for in [9] . we 1.16. DEFINITION will need the [18]. following Let F:K- L be an INS-functor. subcategory U of K is called a PIS-subcategory under products and INS-subobjects. 426 of K, A strictly full if U is closed Recall that each extremal K-monomorphism in an INS-category K embedding [18, 19 , 20]. Furthermore if K is a complete, wellpowered and cowellpowered category, then a strictly full subcategory is epireflective in K if and only if it is closed unde r products and extremal sub- is an objects [9] . Hence we 1.17. PROPOSITION. and wellpowered, is epireflective obtain the Let F:K-L an INS-functor over a category L . Then each cowellpowered I f moreover in PIS-subcategory and in K . then the notions complete, PIS-subcategory L each monomorphism is a kernel, epire flective subcategory are equi- be [18]. valent strictly full subcategory of the category of locally convex spaces is epireflective if and only if it is a PIS-subcategory, i.e. closed under products and subspaces, since the category of locally convex spaces is an INS-category over the category of complex-valued So for instance vector a spaces. Algebraic categories of Z-continuous 2. functors with values in INS- categor ies . Let tors us is or : A a - b the is morphisms, a 2-continuous funcand 2 C Mor K a class of standard notions a category, on called 2-bijective, or Z-continuous, shall denote the full in K . If C is of functorial some Let K be K-object k mapping bijective. 2 K objects tor recall [28,29,30,31]. K-morphisms . all briefly subcategory of all 2-bijective and Z C Mor [ C , S ] is pair (C, L) is called a theory. small category then the A : C - K, where K is an arbitrary if for category, is called a K- class A func- a 5: -algebra if all functors 2-bijective. theory (C, 2) is called algebraic if the inclusion functor Z ( C , S )->. [C, S] is adjoint. Let 2* denote an arbitrary class of theories. A category K is called 2* -algebraic if for all theories (C, L) are A 427 in Z* the inclusion Herewith we functor Z ( C , K) -> [C, K] is adj oint, where Z ( C , K ) denotes the full subcategory of all 5i -algebras in K . Let now (C, Z) be a theory, and let F: K - L be an INS-functor with coadjoint right inverse D . Let furthermore obtain the Hence if A is the same a 5i-algebra method one commutative following in diagram [37] : K , then F C A shows that the functor is a 5i-algebra in L. F Z = FC Z ( C , K) has By an adj oint right inverse induced pointwise by that of F . For the rest of this chapter assume that the base category L is complete, in order to be able to apply characterizations of INS-functors and INS-morphisms given in the preceeding chapter, although the following theorems are valid without any restriction. 2.1. THEOREM ( cp. plete category 1 ) The [37]). Let L . Then we F: K - L be obtain the an following INS-functor over a com- assertions : is functor an INS-functor. Denote by the evaluation functors for c E C , and by (C, ¿) and (D, BII ) be theories. A functor f : D - C is called a morphism of theories if the induced functor [f,S] : [C,S I -> [D, S] preserves algebras. If f : (D ,Y) ->( C , Z) is a theory morphism, and K an arbitrary category, then the canonical functor [ f, K ] : [C, K ] -> [D, K] preserves algebras the corresponding inclusion functors. Let furthermore 428 [7. The restriction [37] functor this notation and denoted by if only if 3 ) The class 2* (vK, vL ) vc is an if there is no an INS-morphism. In particular vK is adjoint is adjoint. pair ( E, E ) of K is particular fK algebraic misunderstanding. With is called obtain we 2 ) The pairand of [ f, K] on Z ( C , K ) inclusion Z*-algebraic if and functors only if L is is INS-morphism. In L* -algebraic for any an of theories. f : ( D , Y) --> ( C, Z) morphism of theories. Then the pair (fK, f L) of algebraic functors is an INS-morphism. In particular an algebraic functor over K is adjoint if and only if the corresponding algebraic functor over L is adjoint. 4) Let As 2.2. a an immediate THEOREM. set, and be a application Let P* be let F : K- L be obtain the following the class o f all theories (C, 2), where I is an INS-functor over a locally presentable cawe tegory L . Th en 1 ) K is P* -algebraic. 2 ) Each P* -algebraic functor over K is adjoint. 3) Each evaluation funct or vc: L (C , K) - K is adjoint. 4 ) Z( C , K) is complete, cocomplete, wellpowered and cowellpower- ed. 5) Z( C, K ) is again an INS-category over a locally presentable tegory. 6) Each theory ( C, Z) defines a functor Z(C,-):Initial(L)-·Initial(Z(C,L)) In particular Z (C ,H) is adjoint for all 429 INS-morphisms H. ca. 2.3. Theorem 2.2 allows REMARK. us to construct in a simple way a lot INS-categories over locally presentable categories. So for instance start with an INS-category over the locally presentable category S , the category of sets: F:K-S. Then take any locally presentable theory (C,¿), as e.g. an algebraic theory in the sense of LAWVERE, a GROTHENDIECK-topology, or more general a limit-cone bearing category in the sense of BASTIANI-EHRESMANN [24]. Then the functor of FZ: Z(C, K)->Z(C, S) locally presentable category 5i ( C , S ) . Now one can continue with this procedure applying theorem 2.2.5. Thus one obtains that the categories of topological, measurable, compactly generated, locally convex, bornological or zero dimensional spaces, groups, rings, sheaves...are P* -algebraic, bicomplete, biwellpowered... is an INS-functor over the Since with K also KOP is locally presentable we get the following category is 2.4. K be an a THEOREM. category. Then Let we obtain the INS-category over following statements : 1) K°p is P* -algebraic. 2 ) Each P* -algebraic functor 3 ) Each evaluation functor 2.5. a locally presentable KOP is adjoint. ( C, K°P ) - KOP is adjoint. over vc:¿ regard algebraic categories of Z-continuous subcategories of INS-categories. Let over INS-category, and since each dual of P* -algebraic ( B A ST I A NI unpublished), an now us THEOREM [37]. Let F:K-L be an INS-functor functors and UC K be a PIS-subcategory. Then the following assertions are valid: 1) 1f (C, 2) is a theory, then 2 (C, U) is a PIS-subcategory o f, Z (C , K) . In particular i f Z (C , K) is complete and biwell powered then Z (C , U) is an epire f l ective subcategory of Z ( C , K ) . 2 ) 1 f L is ( C, Z )-algebraic and if Z (C, L) is cowellpowered, then U is again (C , Z) -algebraic. From this theorem follows for instance that each 430 epireflective subcategory of an INS-category over S reflective subcategory of the category algebraic. 3. Monoidal algebraic categories over is P* of -algebraic, or that each epilocally convex spaces is p*- monoidal INS-categories. theory of monoidal universal algebra over monoidal categories, exactly over S-monoidal categories, is in some way a generaliza- The or more equationally defined universal algebra in the classical sense. The S-monoidal categories, defined by M. PFENDER [35] using ideas of BUDACH-HOEHNKE, are a generalization of monoidal or symmetric monoidal categories in the sense of EILENBERG-KELLY. A S-monoidal theory (C, 0, can) is a small category C equipped with a S-monoidal structure. The S-monoidal algebras are functors from a S-monoidal theory into a S-monoidal category preserving the given S-monoidal structure. Standard examples for this procedure are the monoids in monoidal categories as e.g. the monads over a fixed category, Hopf-algebras, resp. coalgebras in the sense of SWEEDLER, or monoids in the classical sense. In order not to complicate the presentation here by lengthy technical details, I will regard here only monoids over monoidal categories in the usual sense. Everything, which is stated here in the following for these special monoidal algebraic categories, is also valid for arbitrary S-monoidal algebraic categories. By a monoidal functor I always mean a strict monoidal functor. tion of the Let now F : K -> L be has an 3.1. LEMMA. an INS-functor adjoint right-inverse, we F: K - L be Let over a monoidal category. Since F get the an L = ( L , 0, can). Th en there exists INS-functor at least one over a monoidal category monoidal structure on K, such that is a mono’idal In functor. general F becomes a there are a lot of monoidal monoidal functor, as the 431 structures on following examples K, show. such that by ( S, X , can ) the category of sets with the cartesian closed structure defined by the product X . Let now F : K - S be an INS-functor over S . Then the most important monoidal structures on K are the following: 1) K = ( K, II , can ) with the product-monoidal structure lifted by the INS-functor F . In general ( K , rl , can ) is no more cartesian closed as the Denote EXAMPLES. cases K = Top K = Uni f show.(*) or 2 ) Denote by K = ( K, D X , can ) the category structure defined K-obj ect functor by the D : S -> product-monoidal K, K structure on with the monoidal S , and the discrete i.e. Then each functor adjoint, but ( K, D X , can ) is in general not closed monoidal. 3 ) Denote by K = ( K D ,can) the category K together with the «inductive» cartesian product-structure, i.e. k D k’ is the cartesian product on S supplied with the finest K-structure, such that id Fk x Fk’ is an Fmorphism (continuous, uniformly continuous, measurable...( [20])) in each argument. The canonical functorial morphisms « can» are defined as has an in S . Then (K, 0, can) is each coreflective In subcategory it is general tary property, i.e. if again a an not closed monoidal category. So for instance of Top or Unif known if closed INS-category over a monoidality INS-categories over is an INS-heredi- closed monoidal category, is closed monoidal. Furthermore there do rizations of those is closed monoidal. not exist internal characte- cartesian closed categories which again cartesian closed. The most well-known cartesian closed INScategories over S are the categories of compactly generated and of quasiare topological spaces. In the following we assume that K carries any monoidal structure, such that (*) K =( K, il, can.) is cartesian closed iff k sp ecial adjoint functor theorem.) 432 II-, k EK , preserves colimits. ( Apply monoidal functor. Denote by Mon K resp. Mon L the categories of monoids over K resp. over L . With this notation we obtain the followbecomes a ing 3.3. F : ( K , D, can ) -> ( L , D, can ) be a monoidal INS- Then the functor. 1) The 2 ) The defines In Let THEOREM. an following assertions are valid: induced functor Mon F : Mon K -> Mon L pair o f forget ful functors VK : Mon K again -K and an INS- functor. VL : Mon L -> L 1 NS-morphi sm : is particular VK In is is adjoint if and only if VL particular the adj oint right-inverse adjoint. of Mon F is given by Mon J : Mon L - Mon K : with In the prove the one can 3.4. as way for equationally defined algebraic categories following Let K be arbitrary monoidal ,categorw, corresponding forgetful functor. Then LE M MA . be the 1) V 2) same creates an limits. V creates absolute Since coequalizers. lute adjoint functor is monadic coequalizers, we obtain the following 3.5. COROLLARY. an functor . Then (i) and V : Mon K -> K the if and only if it creates abso- Let F: (K, D, can)->(L, D can) be a monoidal INS- following assertions are equivalent : V : Mon K - K is monadic. ( i i ) V : Mon L - L is monadic. 3.6. COROLLARY. functor over a Assume that Let F: (K,D, can)->(L, D, can) be a monoidal INS- monoidal category (L, 0, can) with countable 1 0 L - and - 0 L 1 preserve these 433 coproducts. coproducts. Then V : Mon K- K is monadic. 3.7. The forgetful functor V:Top(R-mod)-R-mod from topological R-modules over a topological ring R into the EXAMPLE. category of tegory of R-modules is R-mod is defined a monoidal INS-functor. The monoidal the ca- structure on product and on Top(R-Mod) by the inductive topology on the tensor product. Since R-mod is closed monoidal, all assumptions of the corollary 3.6 are fulfilled. Hence the forgetful functor by the tensor topological R-algebras into Top (R-mod ) is monadic. the functor « topological tensor algebra» . (k, J.L ’ e) E Mon K. Denote by Lact ( k , K) the category from the category of The coadj oint Let is now of K-objects, be a on which k the left. Let F : ( K , D, ca.) -->(L , 0, can ) monoidal INS-functor. Then F induces With this notation 3.8. acts on THEOREM. Then the induced we a functor obtain the Let F : (K, can) --> (L , can) be a monoidal functor INS- functor. again an Lact F : Lact ( k , K) - Lact ( F k , L ) is INS-functor. 3.9. Let K be INS-category over S , e.g. the category of topological, measurable, compactly generated, zero dimensional or uniform EXAMPLE. an spaces. The category K(Ab) of all abelian groups in K is a closed mo- INS-category over Ab , the category of abelian groups. The monoidal structure is given by the « inductive tensorproduct». The monoids in in K ( Ab ) are just the rings in K . The category Lact ( r, K ( Ab ) ) is the category of all K-modules over the K-ring r. Hence the forgetful functor noidal K(r-mod)-r-mod into the category of all r-modules in Ens is an INSfunctor. Hence the category K (r-mod) is complete, cocomplete, well- powered, cowellpowered, has generators, cogenerators, projectives, injectives and a canonically defined (coequalizer, mono) bicategory structure. But K ( r-mod ) is in general not abelian, since bimorphisms need not to be isomorphisms. 434 4. Algebraic Categories ( in the T H IE B A UD ) of sense over INS-categories. algebraic category [36] includes E ILENBERG-MOORE categories, categories of finite algebras, comma-categor ies... It is defined in a completely natural way by BEN A B 0 u’ s profunctors. I regard here algebraic categories over arbitrary base categories, but restrict myself in the case of the underlying algebraic types to types which are induced by Ens-valued algebraic functors. THIEBAUD’s notion of an 4.1. ALGEBRAIC FUNCTORS AND CATEGORIES defined in his thesis the StrA structure gories over an ( unpublished ) and the semantics arbitrary category A to pair a Sem A of [36]. adjoint the category of comonoids A . This has functors from the category of all in the monoidal category Dist ( A ) of all distributors BENABOU’s THIEBAUD cate- Comon (A) ( = profunctors in adjoint functors induces a monad on (Cat,A). The algebras in the corresponding EILENBERGMOORE category are called algebraic functors, resp. algebraic categories. In the presence of an adjoint (or a coadjoint) the notions of categories algebraic over A and categories monadic (resp. comonadic ) over A coincide. In other words this pair of adjoint functors allows us to associate a category of algebras to an arbitrary functor, in such a way that, if this functor has an adj oint or a coadj oint, then we obtain the category of algebras or coalgebras in the sense of EILENBERG-MOORE. Let us now briefly recall the basic definitions of algebraic functors as well as some of their properties, in particular the stability under pullbacks. We assume that the reader is familiar with bicategories in the terminology) sense of B E N A B O U [25] . 4.1.1. DE FINITION [251. Ens ) - is the over The bicategory given by the following data: set of objects of Dist is the ( sm all rel . to pair of Dist of all distributors (over category Cat Ens); 435 of «all» categories - for A , B E Cat , the category Dist (A , B) of all distributors A -P B category [AoP B , Ens I ; Iet O : A U- B and 9 : B l- C be two distributors. The composition O O Y : A -+-> C is defined pointwise coequalizer in Ens of the followis defined to X be the functor - as ing pair (d0 , d1) of morphisms : These attachments define - for bifunctor A E Cat , the Hom-functor A ( - , - ) is defined distributor - a abbreviation isomorphisms we REMARK are write 00qj OOY (a , c) generated by 4. 1.2. be the identity- 1A : A + A . The coherent natural As to defined in for the an obvious way. equivalence class in with [36]. in I iff there exists a finite chain in B , and elements and such that for all i , visualized by the following commutative 436 diagram : The category Dist (A , A), A E Cat , is the above defined functor « O» the category of all comonoids in adj oint functors functorial Dist (A , A) . We define now the pair of ( StrA , SemA ) . Let f (G, e, d) be are as monoidal category with multiplication. Denote by Comon ( A ) a morphisms. a comonoid on A . In A G- algebra is a particular pair (a, x), x E G ( a, a), such that A morphism f : (a , x) -> (a’ , x’) of G-algebras is a where a E A and A-morphism f : a -> a ’, such that is commutative, i.e. We shall denote underlying forgetful more each by Alg (A , G) functor is denoted comonoid-morphism O: (G , canonical way a functor 437 the category of G-algebras. The by U ( G ) : Alg ( A , G ) -> A . FurtherE, d) -> (G’, E ’ , 8’) defines in a over A . The assignment defines a futictor SemA : Comon (A) -> ( Cat, A). The functor is defined in the fines in following obvious way an two way: Let U : B -> A be functor. Then U de- a distributors : Let O E Dist (A, B ) and 9 E Dist ( B , A ) be two arbitrary 1-cells, i.e. dis- tributors of Dist . Recall that 4Y is if there exist satisfying 2-cells the relations One of the important properties of the bicategory Dist is the fact each functor has a coadjoint, i.e. for any functor U : B -> A most that in Dist the distributor pair of define by the . Dist-coadjoint (left-adjoint) to Y, ( functorial morphisms ) OU : B -l-> A is Dist-caoadjoint Dist-adjoint functors defines StrA (U) the comonoid pair (O U, as OU) of The of functors adjoint pair monad denoted a 8u ) 8U comonad, we can on A generated functors. Structure is adjoint ( StrA , Sem A) to induces semantics. on (Cat, A) a by AIgA . 4.1.4. D E F I NI TI ON . A category algebraic pbisms of algebraic categories 4.1.5. OU : B -l-> A . Since each monad resp. (OUOOU, Dist-adj oint PROPOSITION ( THIEBAUD ) . a to over A are A is AlgA-algebra. the AlgA-morphisms. over an Mor- EXAMPLES. 1) Let G-algebras is G be a algebraic comonoid over on A . Then the category A. 438 Alg (A, G ) of 2 ) Let F : A - C and G : B - C be functors. Then the comma category ( F, G ) is algebraic over A X B . In particular the category (A, a) of objects over a and (a, A) of objects under a, a E A , are algebraic over A . 4.1.6. is monadic . be a [36]. only if U PROPOSITION if and [36 . PROPOSITION pullbacck·diagram in 4.1-8. is algebraic a category and has a over A. Then U coadjoint. Let Cat . Then [36]. PROPOSITION Let U:B -A be if U is algebra!ic Let u : B - A be those limits and colimits which are so is U’. Then U algebraic. creates absolute. 4.2. ALGEBRAIC CATEGORIES OVER INS-CATEGORIES. 4.2.1 . A-obj ects gory of defined as a or category [17]). Let T : A -> Ens be functor. The cate- T-objects T-obj (K) in an arbitrary category over K by the pullback in Cat : K is (cf. DE FI NITION a where Y is the Since 4.2.2. PROPOSITION. algebraic 4.2.3. Yoneda-embedding. with T also [KoP, T] over algebraic if T-obj ( K ) - K One is algebraic algebraic, over we get from 4.1.7 : Ens, then T-obj(K) is K. COROLLARY. category IfA is can is Let T:A - Ens be monadic. Then over K , and in T-obj (K) is a particular again monadic, if and only adjoint. easily prove the following 439 4.2.4. PROPOSITION. T:A-Ens Let be an algebraic functor. Then 1 ) T-obj (K) is complete, if K is complete. 2 ) The forgetful functor T-obj ( K ) - K creates all colimits which the preserved by Let induces 4.2.5. a Yoneda-embedding now F: K - L be an Y: are K ->[K°p , Ens I - INS-functor with coadjoint D . Then F functor THEOREM. Let F:K-L be 1) T-obj F : T-obj (K) -> T-obj (L) an INS-functor. Then is an INS-functor iff T-obj F is fibresmall. This is equivalent to the condition, that there exists up to isomorphisms only a set of structures Ak, k E K, such that (k, Ak ) is a T-obj ( K )-algebra. This for instance is always the case, when K is an INS-category over L and T-obj ( L ) - L is monadic. In the following we assume that T-obj F is always fibresmall. 2 ) The pair of forgetful functors forms an INS-morphism visualized In by particular 6. over UK is monadic if and only if UL COROLLARY ( ERTEL-SHUBERT [6]). is monadic. Let K be Ens , and let T : A - Ens be monadic. Then INS-category T-obj (K)-> K is again an monadic. 4.2.7. COROLLARY [18]. Let K be category L , and let A be an Then the forgetful functor the an INS-category category in the over an arbitrary algebraic of LAWVERE . K is monadic if and only if UK : Alg ( A , K) forgetful functor UL : A lg ( A , L ) - L i s monadic. 440 sense 4.3. CONNECTION BETWEEN THE CATEGORIES OF T-OBJECTS IN K AND PRE-T-OBJECTS IN K. 4.3.1. Let F:K-L be DE FINITION. algebraic functor. The following pullback in Cat : be an i.e. the objects arbitrary « K The 4.3.2. is of -structure » on are the objects of by the with an is defined T-obj (L) it. following propositions PROPOSITION. Pre-T-obj (K) category pre-T-obj (K) INS-functor, and let T:A-Ens an The are categorical routine. forgetful functor INS-functor. an 4.3.3. UPK : Pre-T-obj (K) -> K is algebraic. Up is monadic. The pair (UP, UL ) defines if and only if UL ph i sm. 4.3.4. is monadic PROPOSITION. an INS-mor- Assume that T: A - Ens is monadic. Then T-obj(K) isomorphisms B IR KH O F F-subcategory o f Pre-T-obj (K), i. e. 1) T-obj ( K ) is closed under products in Pre-T»obj ( K). 2) T-obj (K) is closed under extremal monos. 3 ) T-obj ( K) is closed under retracts in Pre-T-obj( K). PROPOSITION. is up to 4.3.5. a Assumption as above. T-obj (K) category o f Pre-T-obj (K) . In particular T-obj (K) is subcategory o f Pre-T-obj (K) for any category K. PROPOSITION. is an an INS-sub- epire flective 4.4. CATEGORIES OF FINITE ALGEBRAS OVER INS-CATEGORIES. this paragraph we have since otherwise the to restrict ourselves following the category of finite sets notions give in Ens . 441 to INS-categories no sense. over In Ens , Denote by Fin (Ens) 4.4.1. F: K - Ens Let DE FINITION. be an INS-functor. The category Fin ( K ) of finite K-objects is defined by the following pullback in Cat : . 4.4.2. PROPOSITION. 1 ) Fin (K) -> Fin (Ens) is an INS- functor. 2 ) Fin (K) -> K is algebraic. The proof of assertion 1 is trivial, whereas 2 follows from the fact that Fin(Ens)-Ens is algebraic [36] and that (*) is a pullback diagram. 4.4.3. ry is is Let T:A-Ens be algebraic functor. The categoT-obj (Fin (K)) is called the category of finite T-objects in K and denoted by Fin (T-obj (K)). In particular Fin (T-ob j (K)) -> Fin (K) an algebraic functor. DEFINITION. 4.4.4. THEOREM. Let T : A -> Ens be be an INS-functor. 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