Herrlich, H., and Strecker, G.E., Category Theory, 2nd. Ed., Sigma Ser. Pure Math. I, Heldermann Verlag, Berlin, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....but the pairs of net morphisms for which the pullback exists are characterized, and the pullback construction is given for these cases. As the notion of a pullback is central for this paper, the section starts with its general definition. For the other concepts from category theory see e.g. [HS79]. Definition4. Pullback. In a category C, the pullback of a pair of morphisms (f i : Y i Z) i=1;2 is another pair of morphisms (g i : X Y i ) i=1;2 such that f 1 ffi g 1 = f 2 ffi g 2 and for every pair of morphisms (g i : X Y i ) i=1;2 with f 1 ffi g 1 = f 2 ffi g 2 , there ....

Horst Herrlich, George E. Strecker. Category Theory. Heldermann Verlag, Berlin, 2nd edition, 1979.

....natural transformation of the functor t to the identity functor, and so that, for each A module M , t t(M) is an isomorphism. This is another way of saying that t is a monocoreflection on ModA , the category of A modules. For elucidation of the categorical terminology the reader is referred to [HS79], x36. In addition, t satisfies the following radical property: t(M=t(M ) 0 for each A module M . In fact, t(M ) is the submodule of M generated by all the submodules which belong to T (Chapter VI, St75] x1) It is well known that T is hereditary if and only if t is left exact, that ....

....is the concept of a perfect Gabriel filter and its relationship to ring extensions B of A which are flat over A and such that the inclusion A Gamma B is an epimorphism of CRng. We assume that the reader is familiar with the concept of an epimorphism in category theory; if not, please refer to [HS79]. It is well known that in CRng epimorphisms need not be surjective. There is a rich theory of epimorphisms in this category, notably in [Sa68] and reprised in [Gl89] Stenstrom gives the following account of this matter (see Chapter XI, Proposition 8 Jorge Martinez Warren W. McGovern 3.4 and ....

H. Herrlich & G. Strecker, Category Theory. Sigma Ser. PM 1, Heldermann Verlag (1979), Berlin.

....G is laterally oe complete; iv) the maximum essential reflection on Ws , denoted s , is the contraction of the maximum essential reflection on W. 1 Contracting Extension Functors to a Monocoreflective Subcategory We begin with some background in category theory. Our basic reference is [HS79]. We assume that the reader is familiar with the very rudiments of category theory. For example, the reader should understand the terms monomorphism and epimorphism, and their adjective cousins monic and epic. All subcategories are assumed to be full. Our domain of discourse is a category A, on ....

....unique A morphism making the diagram in Figure 1 commute, then we say that Phi is a functorial extension operator. The latter definition was introduced in [HM94] where it was shown that if Phi is a functorial extension operator, then, for each A object A, OE A is epic, thus generalizing 36.3 in [HS79] for monoreflections a bit. Conversely, if Phi is an AF, for which each OE A : A Gamma PhiA is always epic, then it is easily seen that Phi is a functorial extension operator. We abbreviate functorial extension operator FEO. An FEO Phi is idempotent if, for each A object A, OE PhiA is an ....

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H. Herrlich & G. E. Strecker, Category Theory, 2nd Ed.; Sigma Ser. Pure Math. 1 (1979), Heldermann Verlag, Berlin.

....re ection morphism, such that, for each R 2 R and each f : G R, there is a unique f : rG R with f r G = f . We say that the functor r : G R is a re ection of G in R. In case every r G is monic in G, R is said to be monore ective, and r a monore ection. If r is a monore ection then (36.3 in [HS79]) each r G is epic. If m : G H is monic we write G H, or if the morphism is signi cant, G m H. Thus, if r : G R is a monore ection, we may write G rG rG. Set r s, for monore ections r and s, if for each G 2 G there is a monomorphism : rG sG such that r G = s G . 2 Anthony W. ....

....i.e. a = A n an implies that (a) B n (an ) c) If A B is Arch epic, A 2 , b is a projection element of B and a 2 A, then a[b] 2 A. Proof. a) is monore ective and, therefore, epire ective (x1) These properties are features of any epire ective subcategory (see 37.2, [HS79]) b) and (c) are 4.1 and 4.6 of [HM97] respectively. 2 The nal preliminary to the proof of Theorem 3.3 is this. Proposition 3.5. If A B is Arch epic, and both A; B 2 , then (A; B) A. Proof. It suces that (A; B) A. Let x 2 (A; B) by Theorem 2.3, x = B n an [b n ] ....

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H. Herrlich & G. E. Strecker, Category Theory, 2nd Ed. Sigma Ser. Pure Math. 1 (1979), Heldermann Verlag, Berlin.

....following fact, which is fairly evident,and, in any event, follows from re ectivity. Proposition 1.2. Let A be an group, and S be a family of subgroups of A, each of which is in D (resp. RU) Then T S 2 D (resp. T S 2 RU) For a general discussion on re ections the reader is referred to [HS79], x36 x37. 2 Unique Hulls Exist We rst present a sketch of an aspect of hull theory for archimedean groups, based, to some extent, on [C73] and greatly simpli ed by free use of the essential closure [C71] We remind the reader that all groups are supposed to be archimedean. Suppose ....

H. Herrlich & G. E. Strecker, Category Theory, 2nd Ed.; Sigma Ser. Pure Math. 1 (1979), Heldermann Verlag, Berlin.

....of types or phrases at different worlds need to be determined. The established formalism for applying the possible worlds ideas to programming language semantics is the formalism of functor categories. Excellent introductions CHAPTER 2. ALGOL LIKE LANGUAGES 13 to Category Theory can be found in [HS79, Gol79, BW90] a quite detailed exposition of functor categories for programming language semantics is given in [Ten91] Only a brief introduction will be given here, to establish notation and to outline the fundamental concepts. 2.3 Functor Category Semantics Let W be some category to be used ....

H. Herrlich and G. E. Strecker. Category Theory. Heldermann Verlag, Berlin, second edition, 1979.

....from settheoretical point of view. There are several ways of justifying it the construction may be explained using either Grothendieck universes, or assumption about existence of several inaccessible cardinals, or using some stratified version of ZFC (as described for example in Chapter II of [9]) The same foundational difficulties arise in the framework of ordinary parchments (cf. 7] 10] as well as parchments [11] In this paper, we shall stay at a naive level, and will not even try to be more precise in this respect. 5 Let us now consider the functor ASrt, as an ....

H. Herrlich, G.E. Strecker. Category Theory, second edition. Heldermann Verlag, Berlin, 1979.

....Cat. Extremal, regular and various other classes of epimorphic functors are characterized and inter related. 1. Introduction The results presented here hinge on a construction that leads to a generalization of the notion of a congruence on a category. According to the usual definition, cf. e.g. [3, 6]) a congruence on a category is an equivalence relation on morphisms. However, it allows only morphisms from the same homset to be related. Such a notion is rather weak. Our programme here is to define generalized congruences so that they capture the essence of functor s operation on its domain ....

....in Cat. The construction of coproducts in Cat is elementary. Hence, a concrete construction of coequalizers presented in section 4 does the job. Yet, despite the fact that cocompleteness of Cat is well known, most authors do not provide a direct and elementary proof of the fact (cf. e.g. [1, 4, 6, 7]) The only place we have found where an elementary construction of # PARTIALLY SUPPORTED BY LOSSED WORKPACKAGE WITHIN THE CRIT 2 PROJECT FUNDED BY ESPRIT AND INCO PROGRAMMES, AND BY THE STATE COMMITTEE FOR SCIENTIFIC RESEARCH GRANT 8 T11C 018 11. c # Marek A. Bednarczyk, Andrzej M. ....

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H. Herrlich and G.E. Strecker. Category Theory. Second edition, Heldermann Verlag, Berlin 1979.

....A f fflffl a B fflffl b A 0 g B 0 (0 00) 3 there exists a unique diagonal A 0 d B making both induced triangles commute. This notion generalizes to sinks a and sources b ; of particular interest will be the case when b is empty. We use the set theoretical foundations of [9], but will talk about collections instead of conglomerates . 1 CONCRETE CATEGORIES For a concrete category hA; Ui over X define the quasi category hA; Ui S of U sieves to be the full subcategory of A Y =U y spanned by all subfunctors of U y images of X objects. U S denotes the restriction ....

Herrlich, H., and Strecker, G. E. Category Theory, 2nd ed. Heldermann Verlag, Berlin, 1979.

....of any d: D U 2 (E) by U 2 and R 2 . Proof. Let us denote by j 1 the unit of the adjunction between R 1 and U 1 and by j 2 the unit of the adjunction between R 2 and U 2 ; then R 2 Delta R 1 is the right adjoint of U 1 Delta U 2 , with unit j = R 2 (j 1 U 2 ) Delta j 2 (see e.g. [18], Proposition 27.8) Let us consider c: C U 1 Delta U 2 (E) and let us show that there exists the pullback of R 2 Delta R 1 (c) along jE . Since C admits extension under R 1 and U 1 , the pullback of R 1 (c) along j 1 U 2 (E) exists and hence, as right adjoints preserve pullbacks, the ....

H. Herrlich and G.E. Strecker. Category Theory, an Introduction. Heldermann Verlag, Berlin, 1979.

....SFPs. We skip the details. 4 A new semantics of EL T In this section we consider a semantics for the necessity modality of EL T , which is based on di erent assumptions about C, M and T . We brie y recall the necessary background about brations and factorization systems, and refer to [1, 9] and [2, 5] for more details. De nition 4.1 (Fibrations) Given p: C B, we say that f 2 C(Y; X) is p cartesian ( for every g 2 C(Z; X) and h 0 2 B(pZ; pY ) s.t. pg = h 0 ; pf) exists unique h 2 C(Z; Y ) s.t. g = h ; f and h 0 = ph p: C B is a bration (over B) for every X 2 C and ....

H. Herrlich and E. Strecker. Category Theory. Heldermann Verlag, 1979.

....where the evaluation functor [A; B] Theta A B is given by ( f) 7 f . Note that naturality of transformations amounts to strong identitivity of the functor precategory. Furthermore, it is easily seen that natural transformations admit a composition as in the case of categories (cf. e.g. [7]) in particular, given as above and functors H : B C and K : D A, we write H = H : HF HG and K = K : FK GK for the natural transformations given by (H) f = H( f ) and ( K ) f = Kf , respectively. REMARK 1.8. The situation is somewhat more complicated in the setting of arbitrary ....

Herrlich, H. and Strecker, G. E.: Category Theory, 2nd ed., Heldermann Verlag, Berlin, 1979.

....mechanism. Section 4 presents normal forms as natural transformations. Section 5 shows the relationship between various database models under the CER model. The concepts of category theory used in the present work are not repeated here; a complete introduction to category theory can be found in [6, 7]. The work presented in this paper is a full development of the ideas reported in [16, 15] 2 Entities and Entity Relationships Our first objective is to provide representations for database entities and relationships in the CER model. In Subsection 2.1 we consider entities; they can be ....

H. Herrlich and G.E. Strecker. Category Theory. Heldermann Verlag, Berlin, 1979.

....from set theoretical point of view. There are several ways of justifying it the construction may be explained using either Grothendieck universes, or assumption about existence of several inaccessible cardinals, or using some stratified version of ZFC (as described for example in Chapter II of [9]) The same foundational difficulties arise in the framework of ordinary parchments (cf. 7] 10] as well as parchments [11] In this paper, we shall stay at a naive level, and will not even try to be more precise in this respect. 3 Let us now consider the functor ASrt, as an ....

H. Herrlich, G.E. Strecker. Category Theory, second edition. Heldermann Verlag, Berlin, 1979.

....of (small) categories, generalized congruences, regular and extremal epimorphisms, transition systems, processes. 1 Introduction Cocompleteness of Cat, the category of small categories, is a part of folklore. Yet, to the best of our knowledge, most of the textbooks on category theory, e.g. [2, 3, 8, 9], do not discuss the subject at all. The only place where we could find a hint in that direction was [1] There, the reader is asked to conduct some abstract nonsense type of reasoning to derive the cocompleteness of Cat as an exercise. The construction of coproducts in Cat is elementary. So, a ....

....and elementary characterization. These are exactly functors injective on morphisms, and hence on objects too. This note attempts to rectify the situation. The results obtained here were made possible only after a successful generalization of the notion of a congruence on a category. Until now, cf. [8], a congruence on a category was defined as an equivalence relation on morphisms from the same homset. Such notion is too weak to characterize functors in the way homomorphisms are characterized by congruences on algebras. Generalized congruences studied here do not suffer from that problem. We ....

Herrlich, H. and G.E. Strecker. Category Theory. Second edition, Heldermann Verlag, Berlin 1979.

....Cat. Extremal, regular and various other classes of epimorphic functors are characterized and inter related. 1. Introduction The results presented here hinge on a construction that leads to a generalization of the notion of a congruence on a category. According to the usual definition, cf. e.g. [3, 6]) a congruence on a category is an equivalence relation on morphisms. However, it allows only morphisms from the same homset to be related. Such a notion is rather weak. Our programme here is to define generalized congruences so that they capture the essence of functor s operation on its domain ....

....colimits in Cat. The construction of coproducts in Cat is elementary. Hence, a concrete construction of coequalizers presented in section 4 does the job. Yet, despite the fact that cocompleteness of Cat is well known, most authors do not provide a direct and elementary proof of the fact (cf. e.g. [1, 4, 6, 7]) The only place we have found where an elementary construction of co equalizers in Cat is presented is [5] There, a two stage construction is presented as a PARTIALLY SUPPORTED BY LOSSED WORKPACKAGE WITHIN THE CRIT 2 PROJECT FUNDED BY ESPRIT AND INCO PROGRAMMES. c fl Marek A. Bednarczyk, ....

[Article contains additional citation context not shown here]

H. Herrlich and G.E. Strecker. Category Theory. Second edition, Heldermann Verlag, Berlin 1979.

....of the supremum of all the corresponding subobjects z y always yields the original x y. Next we consider two special relations on the class of objects of X . 3. 2 Definition Let C ObX Theta ObX be the relation defined by (A; B) 2 C iff every morphism from A to B is a constant morphism (cf. [11], 8.2 8.8) and let K ObX Theta ObX be the relation induced by the Galois connection H op ffi ffi H, i.e. A; B) 2 K iff all essential diagonals m of A and n of B satisfy m n (cf. point (5) at the beginning of Section 2) 3.3 Proposition (1) K C. 2) C K iff X satisfies the ....

H. Herrlich and G. E. Strecker, Category Theory, Heldermann Verlag, Berlin, 2nd ed., 1979.

....partially supported by Macalester College. 3 Research partially funded by the U.S. Office of Naval Research grant N00014 888 K 0455. 1 It is shown that the new Galois connection can be used to obtain the weakly hereditary core of an idempotent closure operator. We use the terminology of [7] throughout. 1. PRELIMINARIES Throughout we assume that X is an ( E , M ) category for sinks, i.e. E is a collection of sinks, and M is a class of X morphisms such that: 0) each of E and M is closed under compositions with isomorphisms. 1) M has (E; M) factorizations (of sinks) i.e. ....

Herrlich, H., and Strecker, G. E. Category Theory, 2nd ed. Heldermann Verlag, Berlin, 1979.

....partially supported by Macalester College. 3 Research partially funded by the U.S. Office of Naval Research grant N00014 888 K 0455. 1 It is shown that the new Galois connection can be used to obtain the weakly hereditary core of an idempotent closure operator. We use the terminology of [7] throughout. 1. PRELIMINARIES Throughout we assume that X is an ( E , M ) category for sinks, i.e. E is a collection of sinks, and M is a class of X morphisms such that: 0) each of E and M is closed under compositions with isomorphisms. 1) M has (E; M) factorizations (of sinks) i.e. ....

Herrlich, H., and Strecker, G. E. Category Theory, 2nd ed. Heldermann Verlag, Berlin, 1979.

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Herrlich, H., and Strecker, G.E., Category Theory, 2nd. Ed., Sigma Ser. Pure Math. I, Heldermann Verlag, Berlin, 1970.

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