Concrete Categories Research Articles

RADICAL AND SEMISIMPLE CLASSES IN CATEGORIES

Abstract In this paper we show that the definition and construction of radical and semisimple classes of associative rings can be interpreted in a general category K in terms of two subclasses of epimorphisms and mono-morphisms. We also provide answers to the following two questions posed by Wiegandt: Which objects should be excluded when defining radical and semisimple classes? Given a concrete category, what should the relationship be between the objects used in defining radical and semisimple classes?

On a product dimension of graphs

The Dushnik-Miller dimension of a partially ordered set (X, <) is defined (see [ 11) as the least number of linear orderings L, ,..., L, of X such that <= nj L,. Equivalently, it is the least number of linearly ordered (Xi, Li) such that (X, <) can be embedded as a spanned subobject into X (Xi, Li) (see [lo]). In a more general setting, we are often encountered with the following situation: We are given a class q of objects of a given nature (a concrete category)-e.g., posets, graphs, digraphs, particular kinds of these, etc.-and a subclass 9 of g such that every C E g can be embedded into a Xi”= 1 Bi with Bi E 9. It is then natural to regard the necessary number of the Bi as a measure of complexity of C, called the dimension (with respect to 59 and 5Y) of c. There is something like a “most canonical” 9 associated with a given V, namely, the class of all subdirectly irreducible objects (to avoid difficulties, let us assume the g in question to consist of finite objects). This is the case with the Dushnik-Miller dimension above if we regard the partial orderings as antireflexive, the linear orderings being the subdirectly irreducibles in the category in question. This is also the case with the dimension of graphs (symmetric graphs without loops) we discuss in this paper. Another example is the dim, studied, e.g., in [ 141: if we consider the category of reflexive partial orderings, the complete system of subdirectly irreducibles consists just of the l-chain and the 2-chain. Similarly, one could consider a dimension of a bipartite graph X as the least n such that X is isomorphic to a subobject of Pi, where P, is the 3-path, more generally, a dimension of a k-chromatic graph as the least n such that X is isomorphic to a subgraph of (P3 @ KkvZ)*, etc. (see [8, 121; for related representations of graphs see also (5, 131). Sometimes, a larger 9 may be more convenient. There are cases where there are many objects which, for given purposes, should be considered as basically simple although they are reducible. For example, representing

Cartesian closed initial completions

We construct cartesian closed extensions of concrete categories with special (topological) properties. As a consequence we find a necessary and sufficient condition for a concrete category to have finitely productive, cartesian closed initial completion. Finally, we exhibit a topological category, not satisfying this condition; this gives a negative answer to the problem of Herrlich and Nel whether each topological category has a cartesian closed topological bull [6]. These results have been announced in [1].

Internal Hom Functors for Polarities

The notion of functionality for an internal horn functor H in a concrete category K was introduced in Banaschewski and Nelson [1], formalizing the condition that the structure on the H(A, B) is "pointwise" structure on sets of functions.

The isomorphisms of the category of uniform spaces and related categories

In this paper we determine the isomorphisms of the categories of uniform spaces, proximity spaces and all full embeddings of the categories of several classes of topological spaces (corresponding to usual separation axioms) into themselves. In the lemmas we investigate the relation of the forgetful functors to concrete categories and to functors between concrete categories, in a general setting. To finish we mention several unsolved problems. Uniform and proximity spaces are not assumed to be separated. Unif denotes the category of uniform spaces, HUnif denotes that of separated uniform spaces. THEOREM. Every isomorphism i of Unif onto itself is of the following form: for each uniform space X there is an isomorphism ix:X~i(X), and for each map f: X~ Y the following diagram commutes: X f.~Y ix iv i(X) ~(f), i(Y)

Cartesian Closed Topological Hulls

It is shown in this paper that if a concrete category $\mathfrak {A}$ admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then $\mathfrak {A}$ admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.

Tensor Products and Bimorphisms

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

Realization of small concrete categories by algebras and injective homomorphisms

Realization of small concrete categories by algebras and injective homomorphisms

Example of a concrete category with no embedding into sets under which every epimorphism is represented by a surjection

Example of a concrete category with no embedding into sets under which every epimorphism is represented by a surjection

A characterization of k-ary algebraic categories

Let k be a regular infinite cardinal number. A primitive (=equationally definable) class of algebras is called k-ary provided the algebras can be described in a specified way by means of operations whose arity is less than k. Such k-ary primitive classes of algebras (resp. k-ary varietal categories) have been characterized categorically by Lawvere [5] in case k = ℵo and by Linton [7], Felscher [1], and others in the general case. In this paper we propose a general definition of k-arity for arbitrary concrete categories, show that this concept is useful at least for algebraic categories, defined below, and coincides with the corresponding concept for varietal categories.

On the quasivariety of convex subsets of affine spaces

By Maleev's theorem a quasivariety of algebras is determined up to rational equivalence by its underlying concrete category (that is category with underlying set functor), so one loses nothing ff one identifies a quasivariety with its underlying concrete category. In this paper we show tha t the concrete category ~ of convex subsets of real affine spaces and restrictions to convex subsets of affine maps is an axiomatic quasivariety, and we describe the algebraic structure of ~. F~rrz OSTERMA~N and Jt)RGV, N SCHMIDT have shown in [4] tha t the concrete category 9 /of real affine spaces and affme maps is a variety, and have described the algebraic structure. Their result is stated in section 1. The axiomatisation of convex subsets of affine spaces as algebras is ~ v e n in section 2 and proved in section 3. The final section is devoted to a general discussion of the quasivariety ~ and the var iety generated by it. I n particular we show tha t !~ has only one nontrivial proper subvariety, namely the variety of semilattices. In the following affine spaces will always be affine spaces over the real numbers R. We remark tha t OSTERMA~N and SCHMIDT proved their result for affine spaces over more general rings.

DEVELOPMENTAL AND EXPERIENTIAL FACTORS IN MAKING WISHES

ADULT ALTRUISTIC GRADE M SD M SD M SD 1 0.65 1.01 1.05 0.99 0.05 0.22 4 1.70 1.10 2.00 0.89 0.35 0.58 7 2.20 0.75 2.55 0.67 0.55 0.86 10 2.75 0.43 2.75 0.44 1.05 0.94 HRC......... 1.50 1.20 2.62 0.81 0.13 0.40 DTS......... 1.70a 0.96 1.20m 0.83 0.00 0.00 * The wish to leave the institution is omitted from this mean. effect (F = 25.17, p < .001), with the other effects and interactions noncontributory. Individual t tests showed that the differences between grades 1-4, 0.95, and 4-7, 0.55, were significant (t = 4.44, p < .001; and t = 2.57, p < .01), while the difference between grades 7-10, 0.20, was not (t = 0.23). The percentages of adult scores by grade were 33.3, 66.7, 86.7, and 90.9, respectively. The percentages of adult wishes in the concrete category by grade level were 27.1, 52.0, 94.1, and 100.0, respectively. The percentages of adult wishes in the abstract category were 54.6, 77.1, 82.9, and 90.0. While there is a trend for grades 1 and 4 to make their abstract wishes more adult than their concrete wishes, grades 7 and 10 make predominantly adult wishes in both abstract and concrete categories. It is evident that child and adult categories were not identical with concrete and abstract, respectively. An overall analysis of variance of altruistic wishes was not feasible because of the extremely small variance of grade 1. An analysis restricted to grades 4, 7, and 10 yielded a significant grade effect (F = 5.65, df = 2/68, p < .01) with grade 10 significantly more altruistic than grade 7 (t = 2.38, p < .025). A nonparametric analysis of the number of Ss making one or more altruistic wishes (1, 6, 7, and 14 for the four grades) yielded a highly significant statistic (X2 = 19.90, df = 3, p < .001); subsequent comparisons of grade 10 with Grades 7 and 4 also showed significant differences (X2 = 4.91, p < .05; X2 = 6.40, p < .02); a comparison of grade 1 with adjacent grades was not feasible because of the extremely low expected frequency per cell. The percentages of altruistic wishes for the four grades were 1.7, 11.7, 22.9, and 38.2. Since altruistic wishes were found to be abstract and the frequency of abstract wishes varied with grade, the relative proportion of altruistic wishes over total abstract wishes was computed, yielding the following percentages, 9.9, 20.0, 27.5, and 42.0. It is legitimate to conclude that altruistic wishes were rare in grade 1, and that the largest increment with respect to number of Ss making wishes of this type, as well as the mean number of such wishes, occurred between grades 7 and 10.

Representations of concrete categories in the category of sets

An example of a concrete category which cannot be isomorphically embedded in the category of sets such that all monomorphisms are represented by injections is specified.

Baer-invariants of algebras

Explicitly this paper deals with functors of the category ,A of associative algebras-not necessarily with identity-over a commutative ring A with identity. The contents of the first part remain valid however in a much more general situation. We have formulated the theory of Baer-invariants in ??1-3 in the language of A-algebras, in order to exhibit their significance in a concrete category and also in order to preserve the continuity with the later sections. The translations into abstract terms does not present any difficulty. It is in particular clear that the definitions and results of ??1-3 apply equally well to any variety (D of groups with multiple operators in the sense of P. J. Higgins (cf. [3]). With every subvariety !3 of (D there is associated a subfunctor and a quotient functor of the identity functor of (D (cf. ?1, see also [5]). The absolute Baerinvariants appear then as their derived functors (?3), the procedure involved being, however, quite different from that in [6]. The principal homological tool is that of a normal short complex (cf. ?2). Relative invariants are obtained by considering pairs of varieties F83 and t32 (cf. ?3). Historically one may take as starting point of the theory the discovery by Hopf (cf. [4]) that if

FOUNDATIONS OF THE THEORY OF CATEGORIES

CONTENTS Introduction § 1. The definition of a category § 2. Isomorphism. Duality. Functors § 3. Functors of several variables § 4. Subcategories. Concrete categories § 5. Invertible maps. Equivalent objects § 6. Monomorphisms and epimorphisms. Subobjects § 7. Null maps, null objects § 8. Normal monomorphisms and epimorphisms § 9. Kernels of maps § 10. Images of maps § 11. Operator objects and maps § 12. Direct and free joins § 13. Simplest properties of direct and free joins § 14. Direct joins in categories with null maps § 15. Discrete direct joins. Categories with summation of maps References