Contravariant Functor Research Articles

The set of multiplicative predifferentials and the rational cohomology algebra of fibre spaces

For a given commutative graded algebra H over a field K of characteristic zero the contravariant functor D n (∵ H) from the category of topological spaces to the category of pointed sets is defined. To a Serre fibration F → E → X the element d(ξ) ∈ D n (X:H). H = H(F: K) , is functorially assigned which determines a new model for ξ. Moreover, the description of the set [ X, Y] 0 is given.

CATEGORY OF VECTOR BUNDLES ON ALGEBRAIC CURVES AND INFINITE DIMENSIONAL GRASSMANNIANS

Equivalence between the following categories is established: 1) A category of arbitrary vector bundles on algebraic curves defined over a field of arbitrary characteristic, and 2) a category of infinite dimensional vector spaces corresponding to certain points of Grassmannians together with their stabilizers. Our contravariant functor between these categories gives a full generalization of the well-known Krichever map, which assigns points of Grassmannians to the geometric data consisting of curves and line bundles. As an application, a solution to the classical problem of Wallenberg-Schur of classifying all commutative algebras consisting of ordinary differential operators is obtained. It is also shown that the KP flows produce all generic vector bundles on arbitrary algebraic curves of genus greater than one.

Relative derived contravariant functors and the cohomology of groups

This article investigates the cohomology of a group relative to a collection of normal subgroups. A group-theoretic description of the first relative cohomology is given and used to deduce: a new five term exact sequence; some Hopf type formulas; and a new version of the Universal Coefficient Theorem.

Dualities for torsion-free abelian groups of finite rank

Two well-known dualities have been very useful in the study of torsionfree abelian groups of finite rank: Warlield duality for locally free groups and Arnold duality for quotient divisible groups [Wa, Ar]. In this note we establish a duality, on classes of torsion-free abelian groups of finite rank, which generalizes both Warlield and Arnold duality. Our results were inspired by a recent paper of Fomin [FOG], who constructed a duality which is also a special case of the one presented here. The basic idea is to write a torsion-free abelian group of finite rank (hereafter, “group”) G as a sum G = G, + G,, where G, is locally free and G, is quotient divisible. A dual for G is then obtained by adding the War-field dual of G,, W(G,), and the Arnold dual of Gz, A(G,) (inside of Hom(G, Q)). The idea of breaking up G into a locally free and a quotient divisible part is not a new one-it was investigated by Murley in [Mu]. As noted by Murley, one easy way to obtain G, and Gz is to take a full free subgroup F of G and write G/F= D 0 R, where D is a divisible and R is a reduced torsion group. Then choose subgroups G, and G2 of G containing F so that G,IF = R and G,/F = D. In general, the G, obtained in this way is decidedly non-unique, even up to quasi-isomorphism. It is therefore somewhat surprising that we can form G* = W(G,) + A(G,) to obtain a group which, up to quasi-equality, is independent of the choice ofG,. Further complexity can be introduced due to the fact that the Warlield and Arnold duals treat free and divisible localizations differently, prompting our definition of PX-groups (Definition 2.1). On the category of PX-groups and quasi-homomorphisms (in which objects are quasi-equality classes), G + G* defines an exact contravariant functor (Theorem 2.13). The PX-group G* has a special property, which was also studied by Murley. On the PX-groups with this special property, the “dualizable PX-groups,” * determines a duality (Theorem 3.3), which generalizes the dualities of Warlield, Arnold, and Fomin (Proposition 3.5). We remark 474 OO21-8693/90 $3.00

On injective rational coefficient systems

LetG be a finite group. By a rational coefficient system forG we mean a contravariant functor from the category of canonical orbits ofG andG-maps into the category of ℚ-vector spaces. In this paper we study injective objects in the category of rational coefficient systems forG.

The fundamental duality of partially ordered sets

The representation of partially ordered sets by subsets of some set such that specified joins (meets) are taken to unions (intersections) suggests two categories, that of partially ordered sets with specified joins and meets, and that of sets equipped with suitable collections of subsets, and adjoint contravariant functors between them. This, in turn, induces a duality including, among several others, the two Stone Dualities and that between spatial locales and sober spaces.

On the scarcity of contravariant left adjunctions

It is shown that any pair of contravariant mutually left adjoint functors between varieties of algebrasC andD factors through the retractions of these categories onto the complete lattices of epimorphs of their respective initial objects, and is thus fairly degenerate. However, some interesting contravariant left adjoint functors do exist between categories of finitely generated (or otherwise restricted) algebras.

The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences

For an Artin Algebra Λ of finite representation type, the category Λ-mod, considered as a ring with several objects, has Krull dimension zero. Contrary, for a wild hereditary Artin Algebra this dimension does not exist. In this paper we show that the Krull dimension of Λ-mod for an Artin Algebra Λ of tame representation type is two. The corresponding Krull-Gabriel filtration by Serre subcategories of the category F of finitely presented contravariant functors on Λ-mod leads to a hierarchy of exact sequences in Λ-mod. Influenced by the functorial approach to almost split sequences by M. Auslander and I. Reiten, we investigate the exact sequences whose corresponding functors become simple in one of the successive quotient categories of the Krull-Gabriel filtration of F.

On Evaluation Subgroups of Generalized Homotopy Groups

AbstractG(A, X) consists of all homotopy classes of cyclic maps from a space A to another space X. If A is an H-cogroup, then G(A, X) is a group. G(A, X) preserves products in the second variable and is a contravariant functor of A from the full subcategory of H-cogroups and maps into the category of abelian groups and homomorphisms. If X is an H-cogroup, then G(X, X) is a ring.

Coherence for categories with group structure: An alternative approach

A category with group structure is a monoidal category with a contravariant functor *, natural isomorphisms I+ A * @ A, I -+ A @A * and coherence axioms for the commutativity of some diagrams. The structure is abelian if the category is symmetric monoidal and additional coherence axioms are satisfied. In a recent paper K.-H. Ulbrich [3] has studied the coherence of these categories and proved that any arrow composed of elementary arrows of the structure depends only upon the formal expression of its domain and codomain: any diagram commutes if the vertices and edges are such expressions and composites, respectively. This result is similar to that of S. Mac Lane for monoidal categories in [2]. We present here an alternative way for the study of the coherence of these categories. This paper was written after we read [3] and became convinced that the interest and applicability of the results could justify a more conceptual treatment of the topic which could make it more accessible. For the case of the categories with group structure we use essentially a “diamond lemma” argument after some necessary transformations: a similar argument was used in [2]. For the symmetric case the coherence result is a corollary of [ 11: we provide only the convenient background. This approach gives some insight into the relation between categories with group structure and compact categories which can be explored more deeply when, and if, the applications require it. We also detail some elementary results and consequences which are related to our treatment. In the last part of the paper we show that a group structure on a monoidal category can be given only by the assignment for any object A of an object 305 0021-8693/83 $3.00

ON LATTICES OF CONTINUOUS FUNCTIONS

A detailed study is made of the contravariant functor D which assigns to each compact Hausdorff space X the bounded distributive lattice DX of all functions on X with values in the unit interval, and of some variants of D. In particular, D is shown to have a left inverse, and a duality for compact Hausdorff spaces is derived from D.

On Halin-lattices in graphs

Halin [2] has shown that primitive sets with respect to a subset A of the vertex set of a connected graph G form a complete lattice (Halin-lattice). In this article special contractions are defined such that pairs ( G, A) and these maps form a category HG and that a contravariant functor exists from HG to the category of complete lattices and lattice homomorphisms. Using this functor it is proved that the lattice homomorphism is a well-quasi ordering in the class of all Halin-lattices of rooted trees.

Conspectus of variable categories

Sheaves in geometry are profitably viewed as variable sets 1131. A non-elementary study of variable sets begins with the analysis of the categories [%‘O’, Set] of small-set-valued contravariant functors on a small category V. The objects U of % are ‘the stages of knowledge’ and the arrows r: V-, U in V give ‘the internal development’ for the sets varying over 55’. Finer analysis of a geometric problem at hand usually discloses a small set Z of arrows in [V”, Set] stable under pulling back. The pair (‘Z’, )3) is called a pullbackstable sketch [2,23 ] (or, when 2 consists of monomorphisms, a site [ 11). A sheaf over (U, 2) is an object X of [Ceop, Set] with the property that, for allf:A + C in Z and all h : A --* X, there exists a unique k : C+ X with kf = h. The full subcategory [Y?“, Set]= of [U”“, Set] consisting of the sheaves satisfies the elementary axioms required on a category in order that it should be a quasip yet this gives too restrictive a notion of variable category, by denying the possible 2-dimensional aspect of its internal development. For example, categories in [‘+Zop, Set] amount to functors X from Y?” to Cat. Most naturally occurring arrows in Cat are only of interest up to isomorphism, so that the condition that X should preserve composition strictly is absurd. What should be considered are the homomorphisms of bicategories [3] from Vop to Cat. When 0 has pullbacks, this then includes the basic category ranging over % which is the category %‘/U at stage U and whose internal development is given by pulling back. Devious contortions do exist for converting the latter basic homomorphism into a functor, however, as this strict approach is developed, further obstacles appear. In particular, problems arise in attempting to force Kan extensions of arrows into such basic

Stable equivalence and rings whose modules are a direct sum of finitely generated modules

The representation theory of a ring Δ has been studied by examining the category of contravariant (additive) functors from the category of finitely generated left Δ-modules to the category of abelian groups. Closely connected with the representation theory of a ring is the study of stable equivalence, which is a relaxing of the notion of Morita equivalence. Here we relate two stably equivalent rings via their respective functor categories and examine left artinian rings with the property that every left Δ-module is a direct sum of finitely generated modules.

Limits of bifunctors into a category

Although covariant and contravariant functors of a small category into an arbitrary category cg, also known as inductive and projective systems, have been widely studied, nothing is known of bifunctors of the product of two small categories into c~. It is the purpose of this paper to study such bifunctors, which we call mixed systems. We shall prove that mixed systems have, in a certain natural manner, two limits, and that there exists a natural morphism from one of these limits into the other. The limits and the natural morphisms are proved to satisfy a certain universal property. We also define a category of mixed systems and investigate some of its properties. In forthcoming papers, we shall take a closer look at mixed systems of sets and of modules. Notations have been made to follow [1] as closely as possible. For results about categories and functors, we refer the reader to [2], [3] and [5]. Results on projective and inductive systems can be found in [4]. In what follows, we shall work within a category cg such that every functor of a small category into c~ has projective and inductive limits.

Natural endomorphisms of Burnside rings

The Burnside ring B ( G ) \mathcal {B}(G) of a finite group G consists of formal differences of finite G-sets. B \mathcal {B} is a contravariant functor from finite groups to commutative rings. We study the natural endomorphisms of this functor, of its extension Q ⊗ B \textbf {Q} \otimes \mathcal {B} to rational scalars, and of its restriction B ↾ Ab \mathcal {B} \upharpoonright {\text {Ab}} to abelian groups. Such endomorphisms are canonically associated to certain operators that assign to each group one of its conjugacy classes of subgroups. Using these operators along with a carefully constructed system of linear congruences defining the image of B ( G ) \mathcal {B}(G) under its canonical embedding in a power of Z, we exhibit a multitude of natural endomorphisms of B \mathcal {B} , we show that only two of them map G-sets to G-sets, and we completely describe all natural endomorphisms of B ↾ Ab \mathcal {B} \upharpoonright {\text {Ab}} .

Duality for contravariant functors on Banach spaces

The following notion of duality is studied: IfG is a contravariant functor on Ban, then\(G^x (X) = Nat(G,(X\widehat{\widehat \otimes }.)')\). We derive the following results: Thex-reflexive functors are exactly the maximal subfunctors ofH(.,A) with reflexiveA, Gx isx-reflexive if and only ifG(I) is a reflexive Banach space,Gx=(Ge)x, andGx(X′)=Ge(X)′ ifX′ has the metric approximation property. the last result has the consequence that for the tensor product of functors\(G\begin{array}{*{20}c} {\widehat \otimes } \\ {Ban} \\ \end{array} (X'\widehat{\widehat \otimes }.) = G_e (X)\) holds, ifX′ has the m. A. P.

Homotopy invariance of contravariant functors acting on smooth manifolds

Homotopy invariance of contravariant functors acting on smooth manifolds

Stable equivalence for some categories with radical square zero

For certain abelian categories with radical square zero, containing artin rings with radical square zero as a special case, we give a way of constructing hereditary abelian categories stably equivalent to them, i.e. such that their categories modulo projectives are equivalent categories. Introduction. Let C be an additive category where idempotents split. Denote as in [2] by Mod C the category of contravariant additive functors from C to abelian groups (called C-modules), and by mod C the full subcategory of finitely presented functors. We shall further assume that each object in C is a finite direct sum of indecomiposable objects in C, and that the endomorphism ring Rc of an indecomposable object C is a division ring modulo its radical, so that a decomposition of an object in C into a finite direct sum of indecomposable objects is unique. Assume also that D = mod C is abelian and that the simple C-modules are finitely presented. All the above conditions on C and D = mod C will be assumed throughout the paper. Important cases where the above assumptions hold are D = mod A, the category of finitely generated (left) modules over a left artin ring A, and the case where D is a dualizing R-variety for a commutative artin ring R (see [4] for definition). Let D/P denote the category D modulo projectives (see [4]). We say that D = mod C and D' = mod C' are (projectively) stably equivalent if D/P and D'/P are equivalent categories. For dualizing R-varieties this coincides with our previous definition of stable equivalence [4]. In [4] we discussed Loewy length for dualizing R-varieties D = mod C, denoted by LL(D), coinciding with the ordinary Loewy length in the case of rings. We showed in [5] that if D = mod C is a dualizing R-variety with LL(D) < 2, there is a hereditary dualizing R-variety stably equivalent to D. In this paper we shall obtain this result in a completely different way. Our method here is interesting because we at the same time get a direct way of constructing a hereditary dualizing R-variety stably equivalent to a dualizing Received by the editors August 16, 1974. AMS (MOS) subject classifications (1970). Primary 16A46, 16A62; Secondary 18A25, 18E10. Copyright i 1975. American Mathematical Society

Valuative Criteria for Families of Vector Bundles on Algebraic Varieties

The object of this paper is to derive separation and completeness properties for the families of vector bundles on a nonsingular projective variety X over a fixed algebraically closed field k. By a vector bundle on X we mean a torsion-free coherent sheaf on X. If Xis a curve, such a sheaf must be locally free; thus this definition corresponds with the usual notion of vector bundle. On higher dimensional varieties, it appears that the category of locally free sheaves is too restrictive; for example, in this category, bundles do not in general have complete flags, whereas in the category of torsion-free sheaves, complete flags always exist (Proposition 1). Let S be an algebraic scheme over k. We define a family of vector bundles on X over S to be a coherent sheaf E on X x S, flat over S, such that for each s e S the induced sheaf E. on P`'(s) is torsion-free. We consider two such families E and E' to be equivalent if there is an invertible sheaf L on S such that E _ E' ( p*(L). Starting from the concept of a family of vector bundles, we are led to define a contravariant functor 'F3@ from the category of algebraic schemes over k to the category of sets, in the following way: If S is an algebraic scheme, let '0B,(S) be the set of equivalence classes of families of vector bundles on X over S. Then if T > S is a morphism of algebraic schemes and E a family of vector bundles over S, (g x X)*(E) will be a family of vector bundles over T. It is natural to ask whether the functor F3SX is representable; that is, is there an algebraic scheme V and a family of vector bundles E over V such that Hom (S, V) -= B,(S) for all S? Such a V would be a natural parameter space for the bundles on X, and E would be a universal family. It turns out that this is too much to expect. First of all, there are too many bundles to be parameterized by an algebraic scheme: we must break the functor up into separate parts corresponding to certain natural invariants, for example, the Hilbert polynomial. Next, we must throw away some