Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 Preliminaries Categories, inclusions and institutions are heavily used in this paper, and this section brie y introduces our notation and terminology for these concepts. The reader is assumed to be familiar with basics of category theory, including limits, colimits, functors, and adjoints [20, 19]. jCj denotes the class of objects of a category C, and C(A; B) denotes the set of morphisms in C from object A to object B. The composition of morphisms is written in diagrammatic order, that is, f ; g : A C is the composition of f : A B with g : B C. Cat denotes the category with small ....

....U : MSpec MSpec denote the functor T ; M, taking modules ( A) to modules ( Th (A) Notice that T is also a right adjoint of M, so that Th is (modulo isomorphism) a re ective and core ective subcategory of MSpec. Since the two categories are equivalent, every categorical property [19] of Th is also a property of MSpec. In particular, pushouts are preserved and re ected by M and T , and MSpec is cocomplete whenever Sign is cocomplete, since Th is cocomplete (by [13] De nition 4. A model m satis es M = A) i m j= Vth(M) in this case, we write m j= M . If h : M ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....paper, and this section brie y introduces our notation and terminology for these concepts. Although it seems to include all the standard examples. 4 2. 1 Category Theory The reader is assumed to be familiar with basics of category theory, including limits, colimits, functors, and adjoints [19, 18]. jCj denotes the class of objects of a category C, and C(A; B) denotes the set of morphisms in C from object A to object B. The composition of morphisms is written in diagrammatic order, that is, f ; g : A C is the composition of f : A B with g : B C. Cat denotes the category with small ....

....U : MSpec MSpec denote the functor T ; M, taking modules ( A) to modules ( Th (A) Notice that T is also a right adjoint of M, so that Th is (modulo isomorphism) a re ective and core ective subcategory of MSpec. Since the two categories are equivalent, every categorical property [18] of Th is also a property of MSpec. In particular, pushouts are preserved and re ected by M and T , and MSpec is cocomplete whenever Sign is cocomplete, since Th is cocomplete (by [13] De nition 11 A model m satis es M = A) i m j= Vth(M ) in this case, we write m j= M . If h : ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....determined morphism h such that the following diagram commutes h # # # # # # # # # # # # # # # # # j # # # # # # Proof. Any (regular epi, mono) factorization is unique in the above sense. This is a standard fact of category theory, cf. Proposition 17.18 in [6]. 2 We have now established that each morphism can be factorized into a regular epimorphism followed by a monomorphism in an essentially unique way. The class of regular epimorphisms as well as the class of monomorphisms are closed under composition. This demonstrates that the categories P and C ....

H. Herrlich and G.E. Strecker. Category Theory. Allyn and Bacon, Boston, 1973.

....2 Preliminaries Category theory, inclusions and institutions are heavily used in the paper. In this section we brie y introduce our notation and terminology. 2. 1 Category Theory The reader is supposed familiar with basics of category theory, such as limits, colimits, adjoint) functors [16, 15]. jCj denotes the class of objects of a category C and C(A; B) denotes the set of morphisms from object A to object B. The composition of morphisms is written in diagrammatic order, that is, f ; g : A C is the composition of f : A B with g : B C. Cat denotes the category of small ....

....U : MSpec MSpec denote the functor T ; M, taking modules ( A) to modules ( Th (A) Notice that T is also a right adjoint of M, so Th is (modulo an isomorphism) a re ective and core ective subcategory of MSpec. Since the two categories are equivalent, every categorical property [15] of Th is a property of MSpec, too. In particular, pushouts are preserved and re ected by M and T , and MSpec is cocomplete whenever Sign is cocomplete (because Th is cocomplete [11] De nition 4. A model m satis es M = A) i m j= Vth(M) As usual, we write m j= M whenever m satis es ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....with basic concepts of both category theory and equational logics. The purpose of this section is to introduce our notations and conventions rather than to rede ne known concepts, though some less frequent notions will be reminded. We suggest the books by MacLane [18] and by Herrlich and Strecker [15] for more detail on category theory. jCj is the class of objects of a category C. By abuse, we often use set theoretic notation, such as P 2 jCj. The composition of morphisms is written in diagrammatic order, that is, if f : A B and g : B C then f ; g : A C. If the source or the target of a ....

....(f i : D(i) Cg i2jIj ; C) and each morphism f : K C, there is an i 2 jIj and a unique morphism f i : K D(i) such that f i ; i = f . 3 Factorization Systems At the author s knowledge, the rst formal de nition of a factorization system of a category was given by Herrlich and Strecker 1 [15] in 1973, and a rst comprehensive study of factorization systems containing di erent equivalent definitions was done by N emeti [20] in 1982. However, the idea to form subobjects by factoring each morphism f as e; m, where e is an epimorphism and m is a monomorphism, seems to go back to ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

.... Research Council of Canada Research Grant A4494 y Research supported by Estonian Science Foundation grant #2975 2 Bulman Fleming and Laan A very general setting for limit preservation is provided by the following well known theorems from Category Theory (see for example Herrlich and Strecker [6], p. 167) Theorem 1.2. For a nitely complete category A and a functor F : A B the following assertions are equivalent: 1. F preserves nite limits. 2. F preserves pullbacks and terminal objects. 3. F preserves pullbacks and nite products. 4. F preserves inverse images and nite products. 5. ....

Herrlich, H. and G. Strecker, \Category Theory", Allyn and Bacon, Boston, 1973.

....that of (weak) inclusion systems [18, 37, 12, 13, 53] except that no factorization properties are assumed; however, the weaker notion is adequate for many purposes. Also, sums and products are not needed for many applications. Inclusive categories can play a similar role to factorization systems [36, 48], but tend to have smoother proofs. The following enriches an institution with inclusions [56] Definition 20 An inclusive institution is an institution with its category of signatures and its Sen functor both inclusive. It is distributive iff its category of signatures is distributive, and is ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....unique (up to isomorphism) by the corresponding pushout property. Moreover, r is injective if r is injective. r Gamma1 d (m ) is a proper subset of r Gamma1 (m ) 1 For the fundamental definitions and results of category theory such as pushouts and their properties we refer to [HS73]. Page 2 iff r deletes at least one node. It has cardinality 1 if r is injective, due to injectivity of r in this case. Single pushout transformations at injective matches can be constructed in two steps: 1. Remove from G all dangling edges, i.e. all edges e 2 GE Gamma mE (LE ) whose source ....

H. Herrlich and G. Strecker, Category Theory, Allyn and Bacon, Rockleigh, New Jersey, 1973.

....we use very elementary language of category theory, because it allows us to unify otherwise cumbersome notions. The definitions of category, morphism and isomorphism are all that we use, and even the most elementary reference, such as [Wal91] will prove far more than adequate. However, we use [HS73] as our standard for notation and terminology. For the complexity theory part, we assume familiarity with the basic notation, terminology and results in the theory of algorithm analysis, as may be found in [CLR90] and NP completeness, as may be found in [GJ79] or [AHU74] 1. The Theory of ....

Herrlich, H. and Strecker, G. E., Category Theory, Allyn and Bacon, 1973.

....the intuition behind these operations. Therefore, categorical notions like (mono) subobjects and factorization systems are not proper for some areas of computing. At the authors knowledge, the first formal definition of a factorization system of a category was given by Herrlich and Strecker 4 [HS73] in 1973, and a first comprehensive study of factorization systems containing different equivalent definitions was done by N emeti [N em82] in 1982. However, the idea to form subobjects by factoring each morphism f as e; m, where e is an epimorphism and m is a monomorphism, seems to go back to ....

....of inclusions systems in Birkhofflike axiomatizability results for generalizations of equational logics, and to professor Joseph Goguen for his comments on previous versions of this paper. 2 Preliminaries The reader is supposed to be familiar with the basics of category theory (e.g. see [Lan71, HS73]) In this section we present our formalism and remind the reader some notions used later in the paper. Calligraphic letters denote categories and functors. If A is a category then jAj denotes its class of objects. The composition of morphisms is written in diagrammatic order, that is, if f : A ....

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Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....that the three properties of satisfaction above are natural and verified by any coalgebraic setting over algebraic signatures. This is our reason for the axiomatization of satisfaction described in Section 4. 2. 2 Inclusion systems Inclusion systems are an alternative of factorization systems [22, 28], which promote the idea of unique factorization. Sometimes they are preferred to factorization systems both because they are more intuitive and because proofs tend to be smoother. Inclusion systems first appeared in [11] in the context of modularization, and were then developed and generalized ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....Now call the sets in ZFU conglomerates, the subconglomerates of U classes and the elements of U sets. Then a category has to have classes of objects and morphisms, while a quasicategory can have arbitrary conglomerates of objects and morphisms. Likewise, there are functors and quasifunctors [HS73]. The existence of a universe U is essentially equivalent to the existence of a strongly inaccessible cardinal. It is impossible to prove the relative consistency of this axiom. This is because we can derive ZFU Con(ZF ) A.1) because U is an inner model of ZF , so ZF can proved to be consistent ....

H. Herrlich, G. Strecker. Category Theory. Allyn and Bacon, Boston, 1973.

....by appropriately selecting equations in the condition of a clause, superposition can be restricted to a complete set of positions. 2 Basic Notions We give only a brief introduction to the basic notions we will need. More elaborate presentations can be found for category theory in [Mac72, HS73] for algebraic specification in [EM85, EGL89] for rewrite systems and orderings in [DJ89] and for horn clauses in [Pad88] 2.1 Preliminaries Sequences. A denotes the free monoid generated by a set A, an a 2 A is called list, string, sequence or tuple. The composition is usually ....

H. Herrlich and G. E. Strecker. Category Theory. Allyn and Bacon, Boston, 1973.

....the intuition behind these operations. Therefore, categorical notions like (mono) subobjects and factorization systems are not proper for some areas of computing. As far as the authors know, the first formal definition of a factorization system of a category was given by Herrlich and Strecker 3 [7] in 1973, and a first comprehensive study of factorization systems containing different equivalent definitions was done by N emeti [12] in 1982. However, the idea to form subobjects by factoring each morphism f as e; m, where e is an epimorphism and m is a monomorphism, seems to go back to ....

....to the University of Reunion. 2 On leave from the Fundamentals of Computer Science, Faculty of Mathematics, University of Bucharest, Romania. 3 They called it hE ; Mi factorizable category. 2 Preliminaries The reader is supposed to be familiar with the basics of category theory (e.g. see [10, 7]) In this section we present our formalism and remind the reader some notions used later in the paper. Calligraphic letters denote categories and functors. If A is a category then jAj denotes its class of objects. The composition of morphisms is written in diagrammatic order, that is, if f : A ....

[Article contains additional citation context not shown here]

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....of a category. Our main goal is to give both definitions and properties in the most general form, allowing the computing scientist to apply them in his (or her) fields of interests as a technical device. The weak inclusion systems represent an analogy to the factorisation systems (e.g. see [6]) which have been used many places in computing science (e.g. see [5, 12, 13] and also the older papers [10, 11] The inclusion systems were first introduced in [4] as a categorical tool to study modularisation. In some cases, including modularisation, they are more useful than the ....

....systems. They may be useful in many categorical approaches to computing, especially to modularisation. In [4] there exists, and it is very used, a particular form of them. 1 On the Weak Inclusion System Definition For all the necessary background from category theory, the reader is reffered to [8, 6] and to [9] Also, the reader who is acquainted with model theory [2] and categorical approaches to it (e.g. see [1, 5, 12] will understand more easily the role of weak inclusion systems. We will denote by jCj the class of objects of a category C, and by kMk the cardinal of a set M . The ....

Horst Herrlich and George Strecker. Category Theory. Allyn and Bacon, 1973.

....two maps f : A C and g : B C, set P = f(a; b) 2 A Theta B j g(a) f(b)g with p 1 : P A and p 2 : P B the usual projection maps, restricted to P . One routinely verifies that P is an R algebra, p 1 and p 2 are well defined, and P satisfies the required universal property (See [11] or [8], for example) Remark 3.1.1. In any category with finite products and equalizers (i.e. difference kernels) the pullback is given by this canonical construction. 9 P Fnan Fnan p 2 Fnan Fnan p 1 Fnan Fnan ## G G G G G G G G G A Theta B Fnan Fnan Fnan Fnan ....

.... P 2 Fnan Fnan Fnan Fnan P Fnan Fnan Fnan Fnan Fnan Fnan Fnan Fnan A Fnan Fnan Fnan Fnan C=I Fnan Fnan Fnan Fnan B Fnan Fnan Fnan Fnan k where each of the small squares is a pullback, which implies the outer square is also a pullback (See [8], pg 147, for example) Hence P 00 in the diagram is equal 13 to P 0 above. Now Lemma 3.1.2 implies that all of the maps in the diagram are onto. Therefore P 0 P is onto, and P is a quotient of k[x 1 ; x n ; y 1 ; ym ] Now suppose T 1 and T 2 are two smooth projective ....

H. Herrlich and G. Strecker. Category Theory, An Introduction. Allyn and Bacon, Boston, 1973.

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H. Herrlich and G. E. Strecker. Category Theory. Allyn and Bacon, 1973.

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