Citations: volume 752 of Lecture Notes in Mathematics

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Michael Barr. *-Autonomous Categories, volume 752 of Lecture Notes in Mathematics. SpringerVerlag, 1979.

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Comonoids In Chu: A Large Cartesian Closed Sibling Of Topological.. - Pratt (2003) (Correct)

....lattices, and categories. This is because the closed sets of a comonoid are the open sets of another comonoid on the same set of points. The participants in this coalgebra workshop would recognize it most readily as a comonoid (A, #, #) in chu, the monoidal category of (bi)extensional Chu spaces [1,4,3,8], where is such a Chu space and # : A# # : A are Chu morphisms satisfying the coassociativity and two counit equations. Compare this with the notion of monoid in a monoidal category , I) as a triple (A, #) where is an object of C and : are morphisms of C satisfying the ....

M. Barr. #-Autonomous categories, volume 752 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

Games and Full Completeness for Multiplicative Linear Logic - Samson Abramsky And (1994) (132 citations) (Correct)

....for Player, A as counterstrategies and e as the payoff function, we see some connection with game theoretic ideas. However, this model is very abstract; in fact it forms a particular case of Chu s very general construction of autonomous categories from symmetric monoidal closed categories [Bar79]. In summary, these models have only rudimentary game theoretic content and hence only a very weak relation with our work. 6.3 Blass game semantics Blass game semantics for Linear Logic is by far the nearest precursor of the present work. While we happily acknowledge its inspiration, we must ....

M. Barr. ?-autonomous categories, volume 752 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

Event-State Duality: The Enriched Case - Pratt (Correct)

....of behavior, independently of whether sequential or concurrent. The basic framework for this view then was complete semilattices, modi ed to cater for con ict by replacing bottom by top. Within a month of writing [1] V. Gupta and I [2] had simpli ed and generalized this framework via Chu spaces [3], which has remained our current view for the past decade [http: chu.stanford.edu ] Yet earlier [4] we had applied categorical enrichment to a uni ed treatment of ordered time, real time, etc. but at that stage of thinking did not have the notion of information as dual to time. What we did ....

....of automating it. Ordinary Chu spaces are a sort of halfway house between universal algebra and category theory. Enriched Chu spaces make the corresponding connection for enriched categories, in the process enriching universal algebra analogously. Ironically the original de nition of Chu spaces [3] was for the enriched case, with ordinary Chu spaces receiving only a passing mention. The rst detailed treatment of ordinary Chu spaces was by Lafont and Streicher, and they were subsequently adopted by Gupta and Pratt [2, 11] for the purpose of modeling behavior at a more fundamental level ....

Barr, M.: -Autonomous categories. Volume 752 of Lecture Notes in Mathematics. Springer-Verlag (1979)

Towards Full Completeness for the Linear Logic of Chu Spaces - Vaughan Pratt Dept (1997) (1 citation) (Correct)

....set of maps (f; f ) A B and T is the relation defined by T ( a; b) f; f ) S(b; f(a) R(a; f (b) This is the case V = Set, k = 2 of the category Chu(V; k) of V enriched Chu spaces over an object k of symmetric closed category V , defined by M. Barr and studied by P. Chu [2]. In this more general setting A and X are objects of V and R : X k is a morphism of V . 7 Now a column T B, as a subset of A Theta B, is a binary relation between A and B, equivalently an A Theta B matrix over 2. Define the a th row of this matrix, denoted T (f;f ) a , to be fb j ....

M. Barr. -Autonomous categories, volume 752 of Lecture Notes in Mathematics. SpringerVerlag, 1979.

In Proceedings of the 10th Annual IEEE Symposium on Logic in .. - Typed Calculus Of (Correct)

....to fi reductions; semantically, fi reduction is absorbed into equality. For a full list of such equivalences, see [12] 4 Categorical Semantics The typed process calculus can be given a semantics in a suitably structured category. Let C be a com pact closed category (a autonomous category [8] in which Omega and O coincide) with countable biproducts and a functor which interprets the exponential of linear logic (i.e. should be a comonad and each A should have a cocommutative comonoid structure [24] Additionally, let C have a strict monoidal endofunctor ffi, and write monunit : ....

M. Barr. -Autonomous Categories, volume 752 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

Presheaf Models Concurrency - Cattani (1999) (35 citations) (Correct)

....object Proof: Just take the initial category, 0, with no objects and no arrows. # It is also immediately seen that the zero object is the unit for the product coproduct bifunctor. Other pseudo functors are definable and they make Prof into what might be called a autonomous bicategory [7]. Definition 4.3.6 If is a bicategory, we write for the opposite bicategory which reverses the direction of the 1 cells but not that of the 2 cells. Definition 4.3.7 We define a tensor and a dualiser in Prof . Tensor: Define# On objects: the product of categories On ....

Michael Barr. #-autonomous categories, volume 752 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1979. With an appendix by Po Hsiang Chu.

Orthocurrence as both Interaction and Observation - Vaughan Pratt Stanford (Correct)

....[Pra91, Pra92c, Pra92b] Shortly thereafter, with our Ph.D. student Vineet Gupta, we found [GP93] that couples yielded a delightfully simple yet complete model of what we had been working towards. Couples go back further than linear logic. In their categorical form they were rst proposed by Barr [Bar79, Bar91], and in the set theoretic form followed here by Lafont and Streicher [LS91] Barr s inspiration for the notion came in turn from work in so called soft analysis arising out of an idea in Mackey s thesis [Mac45] Barr de ned general V enriched couples, whose carrier, cocarrier, and alphabet k are ....

....in turn from work in so called soft analysis arising out of an idea in Mackey s thesis [Mac45] Barr de ned general V enriched couples, whose carrier, cocarrier, and alphabet k are objects of a symmetric monoidal closed category V , forming the category Chu(V; k) studied by Barr s student P. Chu [Bar79, appendix ]. Lafont and Streicher treated ordinary couples, the case V = Set, i.e. the points form simply a set and likewise the states, under the rubric of games. More recently Barwise and Seligman [BS97] have treated ordinary couples for k = 2 under the name of classi cations, with tokens, types, and ....

M. Barr. -Autonomous categories, volume 752 of Lecture Notes in Mathematics. SpringerVerlag, 1979.

Glueing and Orthogonality for Models of Linear Logic - Hyland (2001) (4 citations) (Correct)

....model of classical linear logic has particular interest as if one applies the Girard translation to it one gets the Diller Nahm variant of G odel s Dialectica interpretation [27, 26] 3. 3 Chu s construction Simple self dualization can also be thought of as a special case of Chu s construction [18, 8]. We brie y recall the essentials. Suppose that K 2 C is an object in a symmetric monoidal category C. The category Chu(C;K) is de ned as follows. Objects of Chu(C;K) are pairs (U; X) of objects of C with a map U X K. 13 Maps from U X K to V Y K in Chu(C;K) are ....

P.-H. Chu. -Autonomous categories, chapter Constructing -autonomous categories. Volume 752 of Lecture Notes in Mathematics [6], 1979. Appendix.

Glueing and Orthogonality for Models of Linear Logic - Hyland (2001) (4 citations) (Correct)

....is the category of complete lattices and W preserving maps; abstractly it can be identi ed with the EilenbergMoore category of algebras for the power set monad. W Lat is a model of classical linear logic. Multiplicative structure. The category W Lat was identi ed as autonomous by Barr [6]. The tensor product A B classi es maps A B C preserving suprema in each component; and the linear function space B C is the lattice of all W preserving maps from B to C with the pointwise order. Barr notes explicitly that W Lat is not compact closed. Note, however, that I = so ....

....In general the duality on C d is not particularly noteworthy. However it is an important fact that if C carries enough structure then C d will be a model of classical linear logic. Since (in the presence of a terminal object) one can regard C d as a degenerate form of Chu s construction [6], the result for the multiplicatives and additives should be well known. It still seems worth pointing out just how little is needed to make this work. In particular this construction can be used in situations where the general Chu construction does not have good structure, such as for the ....

M. Barr. -Autonomous categories, volume 752 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

Exact Completions and Toposes - Menni (2000) (5 citations) (Correct)

....can be better understood. 1.1 History The history of topos theory and of regular and exact categories will not be discussed here. But in order to enter into the mood of this section let us mention some early references. For topos theory see [4, 56, 57] and for regular and exact categories see [6]. The contents of the section are divided in two. The first part is a chronological perspective on the events, results and problems that concern us in this thesis. We start with the conception of realizability toposes and then we emphasize the work 5 6 Chapter 1 Introduction intended to ....

....of S. 18 Chapter 2 Regular, exact and lextensive categories In contrast with topological spaces, the underlying set functor does not have a left adjoint. Nevertheless, the functor does preserve finite limits. 2. 3 Regular categories In this section we introduce regular categories [6, 15, 32, 75]. The intuition behind these categories is that a good class of quotients exists and moreover, these quotients are well behaved. Recall that the kernel pair of a map f is the (parallel) pair of maps that form the pullback of f along itself. Definition 2.3.1. A category with finite limits is ....

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M. Barr, P. A. Grillet, and D. H. van Osdol. Exact categories and categories of sheaves, volume 236 of Lecture notes in mathematics. Springer Verlag, 1971.

A Typed Calculus of Synchronous Processes - Gay, Nagarajan (1995) (14 citations) (Correct)

On the Axiomatisation of Boolean Categories with and without.. - Stra�burger (2005) Self-citation (Categories) (Correct)

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Michael Barr. *-Autonomous Categories, volume 752 of Lecture Notes in Mathematics. SpringerVerlag, 1979.

A Characterization Of The Left Exact Categories Whose Exact.. - Menni (1999) (3 citations) Self-citation (Categories) (Correct)

....of the category of topological spaces is locally cartesian closed. In this paper we provide necessary and su#cient conditions on a category with finite limits for its exact completion to be a topos. 2. Regular and exact categories In this section we review regular and exact categories [2, 10, 4]. Definition 1. A category with finite limits is regular if 1. every kernel pair has a coequalizer 2. pullbacks of regular epis are regular epis. It follows that a regular category has stable regular epi mono factorizations. Definition 2. A diagram X e 0 # e 1 # X e ## X ## is called ....

M. Barr, P. A. Grillet, and D. H. van Osdol. Exact categories and categories of sheaves, volume 236 of Lecture notes in mathematics. Springer Verlag, 1971.

On Proof Nets for Multiplicative Linear Logic with Units - Stra�burger, Lamarche (2004) (Correct)