Francis Borceux. Handbook of Categorical Algebra, volume I, II and III. Cambridge University Press, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....f ; t C;B in C , and f C = s A;C ; id C f ; t C;TB ; Ts C;B . It is easy to check that for f : A A we have A g; f B g;y A ;B f B;A B= 0 ;B 0 (in C ) so if T is commutative, they are equal, and define a symmetric monoidal structure on C T . 2. 3 Fibrations Fibrations [Jac99, Bor94b] provide a categorical framework for indexing and substitution. They have been used in Computer Science, for example, to model type theories [Jac99, Cro93] such as polymorphic and dependent types, and to model renaming in models of concurrency [WN95] In this thesis, we will use the dual concept ....

....4.1: Equations for CCS T ( f ) T (A) T (B) as the function which substitutes f (a) for every free name a in a term. Because a presheaf is a special form of an indexed category (where the image of each object is a discrete category, i.e. a set) we can apply the Grothendieck construction (see [Jac99, Bor94b]) to obtain a (discrete) co fibration. Objects are then pairs (A; p) where p is a term with free names in A, and morphisms (A; p) B;q) are functions f : A B such that pf f g = q. The particular structure of the algebra, as a monoid with parallel composition and the nil process, allows us to ....

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Francis Borceux. Handbook of Categorical Algebra, volume 2. Cambridge University Press, 1994.

....some of the mathematical structures which are used in this thesis. We presuppose some fundamental definitions of category theory such as category, functor, natural transformation, co )limit, adjunction, and cartesian closed category, which can be found in any textbook on category theory, such as [ML71, Pie91, BW95, Bor94a]. 2.1 Monoidal Categories Definition 2.1. A monoidal category is a category C with a functor : C C C , called the tensor product, an object I in C , called the tensor unit, and natural isomorphisms a, l and r with components B) B I A satisfying the coherence conditions given ....

....A . The category We will now present the objects and morphisms of the process category Proc, and prove that it is a category. The objects of the category Proc are the objects of A . A morphism p : A B in Proc is given by a diagram in A (i.e. a co span hs; ti : A B in A , see [Bn67] or [Bor94a]) and an object P in P with ar(P) X . In analogy with the motivation in the previous section, we will sometimes refer to P as the process behaviour of p. We will write ar(P) X as X P, and write a morphism in Proc as the diagram Intuitively, the morphisms s and t describe how the ....

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Francis Borceux. Handbook of Categorical Algebra, volume 1. Cambridge University Press, 1994.

....Let v i : S S i be views for i = 1; 2, and assume for simplicity that all data domains are the same; a pushout can be constructed as follows. The set of entity class names is given by the pushout of fA i : E E i in the category of partial orders; this is a standard construction (see e.g. [5]) and gives a partially ordered set whose elements can be seen as equivalence classes of E 1 E 2 . If the pushout cocone is fA i : E i E for i = 1; 2, then E gives the entity classes of the colimiting schema S , and the maps fA i give the corresponding components of the views ....

Francis Borceux. Handbook of Categorical Algebra, volume 1. Cambridge University Press, 1994.

....regard as the natural framework to study quotients of control systems. Besides providing an elegant language to describe the constructions to be presented, category theory also o ers a conceptual methodology for the study of objects, control systems, in this case. We defer the reader to [16] and [3] for further details on the elementary notions of category theory used throughout the paper. The category of control systems, denoted by Con, has as objects control systems as described in De nition 2.1. The morphisms in this category extend the concept of related control systems described by ....

Francis Borceux. Handbook of Categorical Algebra. Cambridge University Press, 1994.

....obtain c = i 1 i(c) u in (S; i 1 i(c) 2 Im(u) so u is epi in Set and hence in Set T . 17 For the remainder of the exposition assume that is a regular cardinal. Recall the de nition of ltered categories (e.g. Borceux [5] De nition 6.4. 2) and accessible functors (e.g. Borceux [6], De nition 5.5.1, where they gure under the name functor with rank ) The standard reference on accessibility is Makkai and Par [9] The next lemma and the following theorem are due to Admek [2] Lemma 7.8. Suppose T is accessible, C; 2 Set T and S C is a subset with jSj . Then ....

.... . Corollary 7.10. T accessible and preserves weak pullbacks = T admits a nal coalgebra. Corollary 7.11. All functors from the inductive class T : X j Id j P j a i2I T i j Y i2I T i j T K j T T; where K is a set and is a cardinal, admit a nal coalgebra. Proof. Combine Borceux [6], Proposition 5.5.3 (for the restricted powerset) Proposition 5.5.6 (for exponents and products) ....

Francis Borceux. Handbook of Categorical Algebra, volume 2. Cambridge University Press, 1994.

....This was rst established (in dual form) by Admek [1] and subsequently translated to a domain theoretic setting by Smyth and Plotkin [11] De nition 5.1. T is op continuous, if T preserves limits of shape op . Remark 5.2. Such limits exist in Set, since Set is complete (see e.g. Borceux [5], Theorem 2.8.1) Theorem 5.3. If T is op continuous, then T admits a nal coalgebra. Proof. Consider the limiting cone Z p 0 p 1 p 2 D D D D D D D D . 1 T1 oo T 2 1 T oo : oo By assumption, T preserves this limit, i.e. TZ Tp 0 z z z z z ....

....that e i = id S . Hence T e T i = id TS , i.e. T i is mono. So it su ces to show that 1 T i = 2 T i, which holds by commutativity of the above diagram. 7 Bounded Functors and Accessible Functors The de nition below is not the standard one, but equivalent to the standard one, see Borceux [5], Proposition 4.5.2. De nition 7.1. G Set T is a set of generators (sog) if for all C 2 Set T , the unique morphism, which lets G in G;f f M M M M M M M M M M M M M G2G;f :G C G C commute for all G 2 G and all f : G C, is epi. 15 Lemma 7.2. Suppose e : C; D; 2 ....

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Francis Borceux. Handbook of Categorical Algebra, volume 1. Cambridge University Press, 1994.

....Preliminaries 25 It is easy to check that for f : A A 0 and g : B B 0 we have A g; f B 0 = f g;y A ;B 0 and f B;A 0 B= f g; y A 0 ;B 0 (in C ) so if T is commutative, they are equal, and define a symmetric monoidal structure on C T . 2. 3 Fibrations Fibrations [Jac99, Bor94b] provide a categorical framework for indexing and substitution. They have been used in Computer Science, for example, to model type theories [Jac99, Cro93] such as polymorphic and dependent types, and to model renaming in models of concurrency [WN95] In this thesis, we will use the dual concept ....

....T CCS ( f ) t) def = tf f g, where ( f f g is substitution of names. Similarly for e T C it is easy to check that substitution is well defined on equivalence classes of terms. Because a functor C Set is a (discrete) co indexed category, we can apply the Grothendieck construction (see e.g. [Bor94b] or [Jac99] to obtain a (discrete) co fibration. We perform the construction for e T C , and use the CCS algebra structure to make the co fibration into a symmetric monoidal functor. For the purpose of this chapter, we will choose finite coproducts in FinSet as A 1 A n = n [ i=1 ....

Francis Borceux. Handbook of Categorical Algebra, volume 2. Cambridge University Press, 1994.

....Preliminaries We introduce some of the categorical structures which are used in this thesis. We presuppose the definitions of category, functor, natural transformation, co )limit, adjunction, and cartesian closed category, which can be found in any textbook on category theory, such as [ML71, Pie91, BW95, Bor94a]. 2.1 Monoidal Categories Definition 2.1. A monoidal category is a category C with a functor : C C C , called the tensor product, an object I in C , called the tensor unit, and natural isomorphisms a, l and r with components a A;B;C : A B) C A (B C) l A : I A A r A ....

....and prove that it is a category. The objects of the category Proc are the objects of A . A morphism p : A B in Proc is given by a diagram Chapter 3. A Categorical Model for Typed Concurrent Programming Languages 42 B t ## A s ## X in A (i.e. a co span hs; ti : A B in A , see [Bn67] or [Bor94a]) and an object P in P with ar(P) X . In analogy with the motivation in the previous section, we will sometimes refer to P as the process behaviour of p. We will write ar(P) X as X P, and write a morphism in Proc as the diagram B t ## A s ## X P Intuitively, the morphisms s and t ....

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Francis Borceux. Handbook of Categorical Algebra, volume 1. Cambridge University Press, 1994.

....of . If y(A; A y (A) is a particular choice of cocartesian liftings of : L M 2 L for every A over L, the assignment A 7 y (A) extends to a functor y : E L E M . A functor obtained in this way is called a co reindexing. We will use the bred terminology freely and refer to [4, 9] regarding further reading on this subject. That the cobration associated to a parameterised signature is indeed a cobration follows from ( id C ) being a cocartesian lifting of : L M 2 L for every C over L. We conclude this section with some properties of cobrations of coalgebras. ....

....sense even when p fails to be a bration. 3 Note that this need neither be a bred nor a cobred adjunction. Indeed, below U will generally not be bred and F will not be cobred. 9 Note that UL , UM are comonadic and recall that E L has equalisers. It follows from the adjoint lifting theorem (see [4], vol.2, corollary 4.5.7) that y has a right adjoint . Second, that each bre E L has limits, follows again from UL being comonadic and E L having equalisers (see [4] vol.2, proposition 4.3.4) That reindexing preserves limits follows from reindexing being right adjoint, see (3) Let us ....

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Francis Borceux. Handbook of Categorical Algebra. Cambridge University Press, 1994.

....N, let L k be a free abelian group and L k i X(k) an epimorphism of abelian groups. Then the epimorphisms L k X(k) for all k induce a natural transformation L k2N k (L k ) X which is certainly itself an epimorphism. Left adjoint functors and coproduct preserve projective objects [Bor94]. Hence the conclusion. We are in position to prove the proposition : Proposition 3.4. For any category C, the equality H gl (C) L (H) Gl(C) holds. Proof. If M is a globular group, we introduce the complex of abelian groups (C gl (M) gl ) defined as follows : C gl 0 (M) M ....

Francis Borceux. Handbook of categorical algebra, volume 50. Cambridge University Press, 1994.

....of the lax colimit : B ; B ] of B in C at, where for each index i 2 jIj, i : B i B ] is the canonical inclusion of categories, and for each index morphism u 2 I(i; j) u : B u ; i ) j is defined by u b = hu; 1 bB u i for each object b 2 jB j j. Lax colimits (see (Borceaux, 1994)) constitute the most relaxed concept of colimit in 2 categories, where diagrams are required to commute up to 2 cells only (rather than ordinary strict equality) Notice that since the Grothendieck construction is a lax colimit of an ordinary (1 )functor, this simply means that the lax cocone of ....

....## # # # # # # # # # # # # # # I K P K ## C at 5 For a better understanding of the structure of Grothendieck institutions we go here for a rather direct proof of this result. Alternatively one may use the general theorem of existence of weighted colimits in enriched categories (Borceaux, 1994) instantiated to the case of lax colimits. gi.tex; 3 02 2000; 9:29; p.7 8 Since I K is a Grothendieck (2 )category, as notational convention, let us assume that u = hS u ; u i for each u 2 I (either index or index morphism) Let ] hS ] i, where ] S ] op K is ....

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Borceaux, F.: 1994, Handbook of Categorical Algebra. Cambridge University Press.

....that B is really a behaviour system, i.e. B(d) D. This follows directly by Corollary 3 pg. 235 of [17] because d is full (since there are no arrows in C out of d except the identity) and faithful as a functor. The 2 case can be treated similarly by using (the dual of) Theorem 6.7. 7 from [2] 5 applied to the C at enriched case, for example. 11 For understanding the behavioural structure of B, we need to explicitate its construction (see Corollary 2 pg. 235 of [17] for the case of ordinary categories) Corollary 12: The structure of the final 2 behaviour system B is given by: ....

Francis Borceaux. Handbook of Categorical Algebra, volume 2. Cambridge University Press, 1994.

....is complete actually preserves all connected limits. It follows that T preserves limits indexed by the cochain op , and thus the nal coalgebra of T may be constructed as the limit of 1 T1 T 2 1 : in the standard manner. 2 Distributors and Continuous Functors Next, we recall from [1] the de nition of a distributor. De nition 5 A distributor (also called a profunctor or bimodule) from a category A to a category B, written : A # B, is a bifunctor : B op A Set: 2 One can de ne a composite of distributors; then small categories, distributors and natural ....

Francis Borceux. Handbook of Categorical Algebra (vol. 1). Cambridge University Press 1994.

....and has been designed as an extension of LO. In order to furnish the adequate dynamic features to a polynomial ring, a traditional view is to consider a sheaf of rings as a ring which continuously varies. In particular, we intend to investigate the classical sheaf representation of a ring ([20]) a process which would allow us to replace the study of an arbitrary commutative ring with unit, such as for instance the ring LO C defined in this thesis, by that of a family of local rings [3] ....

F. Borceaux. Handbook of Categorical Algebra 3. Categories of Sheaves. Cambridge University press, 1989.

....terminal coalgebra and give some examples. We conclude by giving an overview of the general results and possible extensions of the framework. This paper is intended for readers familiar with the basics of category theory. The notions of a symmetric tensor, a bicategory, monads etc. can be found in [2, 3]. 2 Copy categories This section provides a summary of the techniques of [6] which are used subsequently to develop categories of circuits. A copy category is a category with a symmetric tensor ( Omega ; I; a Omega ; u Omega ; c Omega ) and a natural comultiplication Delta Omega : A ....

Francis Borceux. Handbook of Categorical Algebra, volume 1. Cambridge University Press, 1994.

....complex domain (e.g. for adding temporal extension to objects represented as Y terms [A TK93] in such a way that their mathematical properties are preserved. Once this has been achieved, the attention can turn to the interaction between the formalisms in a particular application. Category theory [BARR90, PIER91, BORC94] deals with objects (characterized by their structure) and morphisms (preserving the structure) This immediately expresses the notion of approximation of one structure by another. Nonetheless, one of the advantages of category theory is its ability to model the interactions between categories ....

Francis BORCEUX, Handbook of categorical algebra (2: categories and structures), Cambridge university press, Cambridge (GB), 1994

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Francis Borceux. Handbook of Categorical Algebra, volume I, II and III. Cambridge University Press, 1993.

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Francis Borceux. Handbook of Categorical Algebra, volume I, II and III. Cambridge University Press, 1993.

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Francis Borceux. Handbook of Categorical Algebra 2. Categories and Structures. Cambridge University Press, 1994.

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Francis Borceux. Handbook of Categorical Algebra 1. Basic Category Theory. Cambridge University Press, 1994.

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Francis Borceux. Handbook of Categorical Algebra 2. Categories and Structures. Cambridge University Press, 1994.

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F. Borceaux. Handbook of Categorical Algebra 1. Basic Caterory Theory. Cambridge University Press, 1994.

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Francis Borceux. Handbook of Categorical Algebra, volume 2. Cambridge University Press, 1994.

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Francis Borceux. Handbook of Categorical Algebra, volume 1. Cambridge University Press, 1994.

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