P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....such proof, i.e. that the above functor be faithful. With full completeness, one has the tightest possible connection between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91]; to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89] and to the completeness conjecture in [Gir91a] 3 We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear ....

P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.

.... generally if products and coproducts coincide, so we have biproducts) this gives rise to an interpretation of non determinism: if p; q : A B then we can define their non deterministic combination by (This is the standard derivation of a semi additive structure from biproducts; see [41, 21]) The pairing of f and g will be a disjointly guarded , and hence deterministic, combination of p and q; the type confusion between products and coproducts allows us to compose this with the codiagonal, which removes the tags rendering p and q disjoint, so that p q does indeed represent the ....

.... P i2I p i = S i2I p i Synchronous product: p Theta q = f( a; b) p ) j (a; p ) 2 p; b; q ) 2 qg Restricted relabelling: p[f ] f(b; q[f ] j (a; q) 2 p; fa 3 bg where f : L L is a partial function on the ambient label set, and fa 3 b means f a is defined and equal to b (cf. [21]) In terms of labelled transitions, these combinators are specified as follows: a : p p i p Theta q Gamma q fa 3 b p[f ] Gamma q[f ] Finally, some notation. Given a set X , we write X for the free monoid on X . We shall generally treat the canonical ....

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P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. North-Holland, 1990.

....theory and Category theory. 3.1 Preliminaries Idempotents Let be a category. An idempotent in is an endomorphism e : A A with e = e. Idempotents have been used in semantics [Sco76] to specify datatypes as subdomains of some universal domain . We recall the construction of Split(C ) [FS91] [also known as the Karoubi envelope ] which formally splits the idempotents in into retraction coretraction pairs. The objects of Split(C ) are pairs (A; e) where e : A A is an idempotent in . A morphism is a triple (e; f; e ) A; e) A ) where f : A A satisfies e ffi f ....

P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.

....In terms of trees, of course, X is nothing else than the subtree of X rooted at x. Things are not quite so smooth when one is dealing with base categories other than P. Still a similar notion of a subcomponent rooted at an element of a presheaf can be given in terms of slice categories [37]. Definition 5.1.4 Let C be a small category and let C be an object of C. Define the slice category C C to be the category of objects the arrows f : C C # of domain C and morphisms h : f g the arrows of C, h : C # C ## such that g = hf . It is immediately seen that C C has initial ....

Peter J. Freyd and Andre Scedrov. Categories, allegories, volume 39 of NorthHolland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1990.

....also in abstract heterogeneous relation algebras. In the literature, besides heterogeneous relation algebras also other, weaker formalisations of relation like structures have been investigated. An especially fine grained system of such formalisations are the various kinds of allegories of [ FS90 ] By using their tool box, one may aggregate monster names like locally complete unitary pretabular allegory to describe axiomatisations of carefully selected aspects of relation like structures. Many of these are also studied independently in other sources and under different names. For the ....

.... ) R , b) Q ; R) R ; Q , c) Q u Q 0 ) Q u Q 0 . iii) For all Q : A B and R; R 0 : B C , meet subdistributivity and join distributivity hold: Q ; R u R 0 ) v Q ; R u Q ; R 0 and Q ; R t R 0 ) Q ; R t Q ; R 0 : 1 This may be a class in [ FS90 ] meaning that there, allegories are not restricted to be locally small. The price of this generality, however, is that join, meet, etc. need to be characterised at a more elementary level, while we can introduce these as lattice operators. We therefore sacrifice that generality for the sake of ....

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Peter J. Freyd and Andre Scedrov. Categories, Allegories, volume 39 of NorthHolland Mathematical Library. North-Holland, Amsterdam, 1990.

....object Z, which we rename Z = Y X . For this to be meaningful, all binary products X must rst exist in C. Notice that we ask for exponentials of the form Y X for all X and xed Y , whereas the word exponentiability refers to a property of a particular object X for arbitrary Y . Peter Freyd [FS90] has used the word baseable for the property that we require of , but recognising the meaning of that word out of context depends on already knowing its association with exponents, whereas exponentiating suggests its own meaning more readily. The word exponent seems to have come from the French ....

....a pretopos. The notion of pretopos is the nitary part of Jean Giraud s categorical characterisation of Grothendieck toposes [Joh77, Theorem 0. 45] These are the properties of the category of sets that we require in order to do algebra and symbolic logic in it, for accounts of which see [MR77] FS90] Tay99, Chapter V] In particular we shall show how to construct quotients by equivalence relations using a split coequaliser. For point set topology, related results in both the open and proper cases are to be found in [Bou66, x5.2] except that there an open equivalence relation is by de ....

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Peter Freyd and Andre Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. North-Holland, 1990.

....Verification as well as prototyping techniques are still missing. This sticks a task for the future. I think it is useful for relating our approach to another mathematical models for process behavior where there are known results like coalgebraic specifications ( JR97] flownomials ( S94] upas ([FS90]) or Chu spaces ( Gup94] Pra95] 4.2.3 Real time issues. In order to express real time information one could introduce the standard process type time. Each specification of a length of a time interval is a method in M time;time . An example for such a method is time 20 msec time ....

P. J. Freyd and A. Scedrov. Categories, Allegories, volume 29 of NorthHolland Mathematical Library. North-Holland, Amsterdam, 1990.

....and Category theory. 3.1 Preliminaries Idempotents Let C be a category. An idempotent in C is an endomorphism e : A A with e 2 = e. Idempotents have been used in semantics [Sco76] to specify datatypes as subdomains of some universal domain . We recall the construction of Split(C ) [FS91] (also known as the Karoubi envelope ) which formally splits the idempotents in C into retraction coretraction pairs. The objects of Split(C ) are pairs (A; e) where e : A A is an idempotent in C . A morphism is a triple (e; f; e 0 ) A; e) A 0 ; e 0 ) where f : A A 0 ....

P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.

....cut free such proof, i.e. that the above functor be faithful. With full completeness, one has the tightest possible connection between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91]; to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89] and to the completeness conjecture in [Gir91a] We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear ....

P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.

....cut free such proof, i.e. that the above functor be faithful. With full completeness, one has the tightest possible connection between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91]; to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89] and to the completeness conjecture in [Gir91a] We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear ....

P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.

....of interaction and presents an internal language based on the typed process theory in [6] But as yet I havn t worked out the proper treatment of recursion. 2 The Axioms We require a category C satisfying the following axioms: Axiom 1 C is semi additive or half additive in the terminology of [5]. Zeros of a hom set C(A; B) will be denoted by nil A B in keeping with process theoretic notation. Axiom 2 Addition on the hom sets is idempotent. This gives a partial ordering on the arrows of the homsets, given f : A B and g : A B we say that: f g ( f g = g joins are given by ....

....that (a; b) f 0 (a;b) f 0 . Lemma 4 For all arrows f : A B, flf = A ; B ) f . Proof: flf = id flA ; flf ; id flB October 1, 1996 14 : 23 DRAFT 6 2. 1 Bi Products It is well known that if a semi additive category has products then it has co products and that they coincide [5]. Given a 2 AC(A) and b 2 AC(B) what can be said about the atomic co reflexives of A Phi B Proposition 6 Suppose a 2 AC(A) then the arrow a l = Gamma id A nil Delta ; a; hid A ; nili 2 AC(A Phi B) 4 similarly if b 2 AC(B) then b r = Gamma nil id B Delta ; b; hnil; id B i 2 AC(A ....

Peter J. Freyd and Andr'e Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. North-Holland, 1990.

.... than those approaches since we draw heavily on additional background in algebraic graph rewriting (see [EKL90] for a tutorial overview) The part of the structure of relation algebra that can readily be exploited in graphical calculi is that of a unitary pretabular allegory (UPA, introduced in [FS90]) Allegories are a generalisation of categories to cope with relation like structures; we shall not need any allegory theory in this paper, but only refer to it for comparison with one of the main streams of related work in the literature. In [BH94] an approach to transformations of expressions ....

....proofs, but concentrate on giving ample motivation and at least a few examples. I gratefully acknowledge the comments of an anonymous referee. 2 Type and Relation Terms The structure we are going to exploit in our diagram proofs is that of a locally complete unitary pretabular allegory (LCUPA) [FS90], which is essentially an abstract relation algebra in the sense of [SS93] without negation) equipped with all direct products (which are understood to be formed in the underlying category of total functions throughout this paper) For improving understandability we shall use the more widespread ....

Peter J. Freyd and Andre Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. North-Holland, Amsterdam, 1990.

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P. J. Freyd and A. Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier Science Publishers, 1991.

No context found.

Peter J. Freyd and Andre Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier, 1990.

No context found.

Peter Freyd and Andre Scedrov. Categories and Allegories, volume 39 of North Holland Mathematical Library. North-Holland, 1990.

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P. J. Freyd and A. Scedrov. Categories, Allegories, volume 29 of North-Holland Mathematical Library. North-Holland, Amsterdam, 1990.

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Peter J. Freyd and Andre Scedrov. Categories, Allegories, volume 39 of North-Holland Mathematical Library. Elsevier, 1990.

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