F. Borceux, Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge University Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the tools we need for the study of the fundamental category can in fact be applied to at least all small categories. These are the notions of categories of fractions, of left and right calculi of fractions and of pure systems. The first two notions are well known in the category theory literature [6, 1] and were already applied to the analysis of fundamental categories in [17, 10] The new notion in this paper is that of pure systems yielding far more satisfactory applications. 3.1. Categories of fractions. In the sequel, we will only consider small categories (most of the results would still ....

....far more satisfactory applications. 3.1. Categories of fractions. In the sequel, we will only consider small categories (most of the results would still hold with locally small categories [14] but we do not need these in the applications to the fundamental category) Definition and lemma 1. [1] Let C be a category. 1) A subset c Mot(C) is called a system of morphisms of C if (i) Vx object of C, Idx (ii) Va:x x ,a2:x x ,a2oa . In other words, the objects of C together with form a wide subcategory of C. 2) Given a system of morphisms in C, there is, up to isomorphism of ....

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F. Borceux, Handbook of categorical algebra I: Basic category theory, Cambridge University Press, 1994.

....system where the arrows will be the abstract epis and the A arrows the abstract monics. This means that G and A will appear as lluf subcategories of qk, and every qk morphism will factorize as a composite of an morphism followed by an A morphism, orthogonal to each other in the usual sense [3, 1]. The requirements induce the definition I AI Notation. An arrow s, which is a triple (X, g, a , will usually be written in the form (g a) The components g G and a A will sometimes be called guard and action, respectively. Conversely, given an arrow f q, we ll denote by f a ....

....with d D(X, Y) in G and by precomposing in A. The category = D is now defined C A s (K, L) fx (K, X) x (X, L) CD If the I arrows support the right calculus of fractions in ( and the left calculus of fractions in A, the coends can be simplified by the usual equivalence relations [4, 1]. In any case, the composition follows from the universal property of the coends, and this general construction yields the following universal property of Proposition 2. Given the categories over the same object class q, the category q is universal for categories C, given with a ....

BORCEUX, F. Handbook of Categorical Algebra 1: Basic Category Theory, vol. 50 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 1994.

....transformation from # U to #U . The heuristic picture here is: SU Theorem 2.4 V 2 categories, V 2 functors, V 2 natural form a 3 category called V 2 Cat. Proof Recall that a 3 category is a category enriched over 2 Cat. This is expanded in terms of axioms in [6]. Our objects are V 2 categories. There are two parts. In part 1 we show that for every pair we have a 2 category made up of V 2 functors, V 2 natural modifications. For now then are the 0 cells, transformations are the 1 cells, and are the 2 cells as in the following ....

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

....holds by saturation. 2) This is an instance of the saturation condition: the left hand side is m 1 k 1 ) Laxity implies the left to right inequality. For the converse, we must see that the domains of definition of both sides agree. By saturation, the Orthogonal to monos (in M) See [Bor94] for this property in the context of factorisation systems. 9 restriction of Dm k n to M is the same as the preimage of Dm 1 n ( 1 ) which formulated with variables reads as: y 1 ; y m ; x 1 ; x k ; z 1 ; z n ) 2 Dm k n (x 1 ; x k ) 2 D k (y 1 ; y m k (x 1 ; ....

.... limit diagram) and consider the corresponding factorisation of the original tent through it: dx ) S S S S S S S S S S S S S S S D y (Tf) F dy A A A A A Call a morphism f : x y an inclusion when f : X Y is an isomorphism (in B ) Recall from [Bor94] the notion of factorisation system. The above diagram and Proposition 5.12 show the following: 5.13. Proposition. Any morphism f : x y of partial algebras factors (uniquely) as an inclusion followed by a Kleene morphism. The pair of classes of morphisms (inlcusions Kleene morphisms) ....

F. Borceux. Handbook of Categorical Algebra I: Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its applications. Cambridge University Press, 1994.

....t t t t c S E E E where the square is a pullback. Horizontal composition of 2 cells is clearly (canonically) induced by that of morphisms, while the vertical composition is inherited from B. 2. 2 The bicategory of relations Rel(B) We now assume a stable factorisation system (E ; M) on B [Bor94]. In this context we can de ne the bicategory of relations Rel(B) as follows: which is a M arrow into X Y . We refer to such a morphism as a relation from X to Y , which we write R: X 6 Y . the spans, commuting with the domain and codomain morphisms as in Spn(B) The ....

F. Borceux. Handbook of Categorical Algebra I: Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its applications. Cambridge University Press, 1994.

....A, Cpo(PA) Cpo] and Cpo(PA) Cpo v ] v . Proof: Implicit in [Ros80, Chapter III] 11.2 Orthogonality A crucial role in the representation theorem below is played by a notion of orthogonality with respect to cones, generalising that of orthogonality with respect to maps (see [Bor94a] and the sheaf condition for a cover in a Grothendieck topology (see [Bor94b, LM92] Convention. In the sequel all cones range over small diagrams. An object is said to be orthogonal to a cone whenever it believes (or, as Lawvere will put it, perceives) that the cone is colimiting. Formally: ....

F. Borceux. Handbook of Categorical Algebra I: Basic Category Theory. Cambridge University Press, 1994.

....R : AB # L and S : B C # L: R # S) a, c) # R(a, b) # S(b, c) b # B . When L = 0, 1 , this agrees with the familiar composition of relations. The authors remark that this operation is associative i# L is join infinitely distributive (JID) also called a locale in [Bor94]. L. Moss, in [Mos99] considers the following subfunctor of R ( Q(X) f : X # R supp(f) finite, # x#X f(x) 1 . Coalgebras of this functor are stochastic transition systems( Mos99] dVR99] Moss invokes the Row Column theorem for R , which is to say that R is (m, ....

F. Borceux, Handbook of categorical algebra 1: basic category theory, Cambridge University Press, 1994.

....j 00 X : G(j 0 X ) ffi j F 0 (X) 5. The existence of terminal and cofree coalgebras It is well known that terminal coalgebras exist for bounded type functors. Boundedness of F guarantees the existence of a set of generators, so an application of the special adjoint functor theorem (see [Bor94]) would guarantee the existence of the terminal coalgebra. In [GS99] we have given a direct construction. As a consequence of our main theorem 4.7, however, we even know how the terminal coalgebra arises from the terminal automaton. If Q is the terminal automaton, as described in subsection ....

F. Borceux, Handbook of categorical algebra 1: basic category theory, Cambridge University Press, 1994.

....the category Set F . 2. If the homomorphism f is a monomorphism (i.e. injective) then it is mono in the category Set F . If the functor F preserves weak pullbacks then the converse is also true: if f is mono then it is injective. Proof: We use the following categorical characterization of epi s [Bor94][Proposition 2.5.6] Let C be an arbitrary category. An arrow a : A B in C is epi if and only if the following diagram is a pushout in C: A a fflffl a B fflffl 1B B 1B B: By Theorem 4.5, the forgetful functor U : Set F Set creates colimits and hence pushouts. Moreover it is ....

....the following. Theorem 10.3 Any functor F for which a set of generators exists, has a final F system. 2 For all bounded functors (Definition 6.7) a set of generators exists. Theorem 10.4 For every bounded functor F , a set of generators, and hence a final F system, exists. 5 See [Bor94][Proposition 4.5.2] for two equivalent characterizations of this notion. 31 Proof: Let V be a set such that for any system (S; ff S ) and any subsystem hsi = T ; fi) of (S; ff S ) T can be embedded in V . The following is a set of generators for F : f(U; fl) j U V and fl : U F (U)g: For ....

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F. Borceux. Handbook of categorical algebra 1: basic category theory, volume 50 of Encyclopedia of mathematics and its applications. Cambridge University Press, 1994.

....Worrell shows in [Wor98] that the product of a family (A i ) i2I of coalgebras exists, provided that the functor F is bounded and preserves weak pullbacks . We shall define and study these conditions in the later sections. A careful analysis of his proof, which mainly rest on a result in [Bor94], shows that preservation of weak pullbacks is not really required. Putting together all necessary ingredients from the proof in [Bor94] becomes extremely complicated and involved. In [GS99b] see also page 48, we offer an elementary proof of this result, of which the following is the main ....

.... preserves weak pullbacks . We shall define and study these conditions in the later sections. A careful analysis of his proof, which mainly rest on a result in [Bor94] shows that preservation of weak pullbacks is not really required. Putting together all necessary ingredients from the proof in [Bor94] becomes extremely complicated and involved. In [GS99b] see also page 48, we offer an elementary proof of this result, of which the following is the main observation: Theorem 6.2 ( GS99b] Let the product A of a family (A i ) i2I of coalgebras exist, and let B i A i for each i 2 I. Then the ....

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F. Borceux, Handbook of categorical algebra 1: basic category theory, Cambrigde University Press, 1994.

....sometimes, following Mac Lane, it is just assumed that at least one universe exists. Typical discussions can be found in McLarty s book Elementary Categories, Elementary Toposes [25] and Borceux s volume entitled Basic Category Theory in the Encyclopedia of Mathematics and its Applications [2]. Universe existence axioms are introduced, in part, as an alternative to proper classes. In systems like HOL and Isabelle, classes correspond to unary predicates in the metalogic [27, Section 3.1] The availability of such predicates weakens the motivation for these axioms, but maybe they can ....

F. Borceux, editor. Handbook of categorical algebra: Basic category theory. Cambridge University Press, 1994.

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

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Borceux, F.: Handbook of Categorical Algebra 1: Basic Category Theory. Volume 53 of Encyclopedia of Mathematics and its Applictions. Cambridge University Press (1994).

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F. Borceux. Handbook of Categorical Algebra 1: Basic Category Theory, volume 53 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.

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Borceux F., Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge U. P., Cambridge, 1994.

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Borceux F., Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge U. P., Cambridge, 1994.

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