M. Barr and C. Wells, Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 275, Springer-Verlag, Berlin (1985).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and procedure calls are mediated through operations on data. This paper shows the relevance of the CPS representation for analysis and assessment of inference systems. Monads and triples. Originally introduced by Moggi [8] in computer science, monads or triples, a notion from category theory [2], were shown to generalise the continuation passing style transformation. Monads can model a wide variety of features, including continuations, state, exceptions, input output, non determinism, and parallelism [16, 17] The monadic style essentially distinguishes between values and computations ; ....

....its multi faceted forms constitutes the major part of the companion paper covering the NXP architecture. 3.2. Categorical background. This section defines the basic notions from category theory that we need in the formalisation of the NXP architecture in monadic style. Readers are referred to [2] for a comprehensive presentation of categories and triples. Let C be a category, we denote by Obj(C) the objects of C and by Hom(A,B) the set of arrows with source object A and target object B. Definition 3.2.1. If C and D are categories, a functor F : C D is a map for which: If f : A ....

Michael Barr and Charles Wells, Toposes, triples and theories, Grundlehren der mathematischen Wissenschaften, vol. 278, New York, 1985, A list of corrections and additions is maintained in [3].

.... coalgebras from the programming perspective and to the categorical approach to functional programming in general, we refer to [Fok92, BdM97] The recursion and corecursion schemes used in the paper are described in [UV99, UVP01] The classic category theory texts treating (co)monads are [Man76, BW84] Throughout the paper, we work in one base category C about which we do not make any specific assumptions other than the existence of the particular coproducts, initial algebras etc. that we name. The category Set of sets and set theoretic functions is always a possible choice for C . The ....

M. Barr and C. Wells. Toposes, Triples and Theories, vol. 278 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1984.

....a left ajoint (t ) 2) The forgetful functor from modules to abelian algebras creates all limits and colimts, and the category of modules is an abelian sub category of the category of abelian algebras. Proof. If nothing else, Part 1) follows from Freyd s adjoint functor theorem, [1], x1.9. Part 2) follows from the de nitions. Proof: We will write t for the left adjoint to the forgetful functor from modules to abelian algebras; that is t = t 0 ) We note at this point that the category of simplicial modules is the category of simplicial objects in ....

M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der Mathematischen Wissenschaften 278, Springer-Verlag, Berlin 1985.

....the forgetful functor A T A S has a left adjoint T S ( characterized by T S S(M) T (M) 3) If S T is a morphism of operads and A is a T algebra, the functor M T A M S A has left adjoint T S ( characterized by T S (A S M) A T M: Proof. All three follow from [2]. The argument is a modi cation of the argument given for Proposition 1.10 ANDR E QUILLEN (CO )HOMOLOGY 11 2. Derivations, di erentials, and (co )homology Andr e Quillen cohomology is obtained as the right derived functors of derivations. We introduce and explore the relevant concepts. If A ....

M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der Mathematischen Wissenschaften 278, Springer-Verlag, Berlin 1985.

....is given in Chapter 5. 14 Sometimes called language or vocabulary in classical logic textbooks. 15 As opposed to to the syntactic perspective that regards terms as tree like syntactic constructs. 16 Not to be confused to the Computing concept of module 6 nicely) the theory of sketches [4], and the recently developed theory of abstract algebraic institutions [89, 91] However, no uniform proof theory has previously been developed for all these equational logics. It could be argued that, at least for computation, the proof theory is more important than the model theory. In ....

Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer, 1984. Grundlehren der mathematischen Wissenschafter, Volume 278.

....tensor algebra on X. The category S T inherits a topological model category structure from S U . There are some technical difficulties in proving this fact. The first is that one must show that S T actually has colimits. This actually a general fact about categories of algebras over a monad ([2]) but we d like to highlight one point. It s enough to prove the existence of coproducts and reflexive coequalizers. Coproducts are simple. A reflexive pair of maps is one that can be made to look like the bottom of a simplicial object; that is, one has maps f; g : X Y and one can choose a ....

M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der Mathematischen Wissenschaften 278, Springer-Verlag, Berlin 1985.

.... due to the comprehensive development of categorical universal algebra; without any claim of completeness, I mention the so called Lawvere algebraic theories, either in classical form [33] or in monadic form [34] although neither of these fits order sorted algebra nicely) the theory of sketches [2], and the recently developed theory of abstract algebraic institutions [37, 38] However, no uniform proof theory has previously been developed for all these equational logics. It could be argued that, at least for computation, the proof theory is more important than the model theory. In ....

Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer, 1984. Grundlehren der mathematischen Wissenschafter, Volume 278.

....is also quite short. We de ne application as a function that takes a term t and the body b of a lambda abstraction, and replaces every occurrence of Zero (the nameless variable bound by the abstraction) in b by t : y These equations are part of the statement that distT is a distributive law (Barr Wells, 1984) between the monads on Term and Incr . de Bruijn notation 11 apply : Term a Term (Incr a) Term a apply t = joinT . mapT (subst t . mapI Var) The function mapT (subst t mapI Var) returns an element of Term (Term a) a term of terms. The function joinT attens such elements into ....

Barr, Michael, & Wells, Charles. (1984). Toposes, triples and theories. Grundlehren der Mathematischen Wissenschaften, no. 278. New York: Springer.

....conclusion is a unique existentially quantified such conjunction. COS( is the institution of coherent order sorted signatures, algebras and theories introduced by Goguen and Meseguer [GM92] with conditional sort constraints [GJM85, Yan93] LESKETCH, the institution of left exact sketches [Gra87, BW85] has graphs as signatures, which are interpreted in the category of sets. Sentences allow to state commutativity of diagrams and properties of products (and other limits in the sense of category theory) In general, the choice of the institution determines whether we just have total operations ....

....n : A Gamma B 1 Theta Delta Delta Delta Theta B n denote the unique morphism h with A h Gamma B 1 Theta Delta Delta Delta ThetaB n i Gamma Gamma B i = A h i Gamma Gamma B i for i = 1; n. 3. 7 Left exact sketches The institution LESKETCH (left exact sketches, see [Gra87, BW85]) has signatures Sigma = G; U ) where G = V; E; E start Gamma Gamma Gamma V; E end Gamma Gamma V ) is a (finite) directed graph (with the possibility of multiple edges between two nodes) and U is a map assigning to each vertex (or object, to keep close to categorical terminology) a in ....

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M. Barr, C. Wells. Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 278. Springer Verlag, 1985.

.... with conditional existence equations: Here, a strong equation t 1 s = t 2 is mapped to (t 1 e = t 1 Gamma t 1 e = t 2 ) t 2 e = t 2 Gamma t 1 e = t 2 ) A more complex example is the conjunctive simple map of institutions from the institution of set valued left exact sketches (see [4]) to the institution of limit theories (an extension of Horn theories with unique existential quanfitication in the conclusion, see [9] which is described in [17] A diagram over a sketch is translated to a set of equations, while a cone over a sketch is translated to the statement that the cone ....

.... Gamma Fnan Gamma Fnan Gamma Fnan Gamma Fnan Gamma Fnan Gamma Fnan Fnan Gamma Fnan Gamma Fnan Fnan Fnan j J Fnan Fnan OO That is, F can borrow those sentences from J which do not lead to new theories. For example, if we take F to be Lawvere theories or FP sketches over Set [4] and map them to their equational theories (therefore we have to equip equational theories with some notion of derived signature morphism) we can consider usual equations as new axioms. 3.4 The Model Class Monad and Semi Maps of Institutions In the previous subsection, we generated sentences ....

M. Barr, C. Wells. Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 278. Springer Verlag, 1985.

....to study relationships between local and global phenomena, for example, in algebraic geometry, differential geometry, and even logic; the subject has also been developed in an abstract form using category theory. The theory of topoi, originally developed by Lawvere and Tierney (see [33] 26] [2]) is perhaps the most exciting development in this respect. An interesting topic for future research is to see what the theory of topoi can tell us about concurrency. For example, one should be able to reason about a system using the internal intuitionistic logic of the corresponding topos of ....

Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer, 1984. Grundlehren der mathematischen Wissenschafter, Volume 278.

....Unfortunately, no existing text is ideal for computing scientists, but perhaps that by Goldblatt [36] comes closest. The classic text by Mac Lane [47] is warmly recommended for those with sufficient mathematics background, and Herrlich and Strecker s book [39] is admirably thorough; see also [2] and [45] The paper [22] gives a relatively concrete and self contained account of some basic category theory for computing scientists, using theories, equations, and unification as motivation, and many examples from that paper are used here. 1 As far as I know, the first such attempt is my own ....

....profound generalisation of the idea that a theory is a category appears in the topos notion developed by Lawvere, Tierney, and others. In a sense, this notion captures the essence of set theory. It also has surprising relationships to algebraic geometry, computing science, and intuitionistic logic [36, 2, 42]. 9 Discussion The traditional view of foundations requires giving a system of axioms, preferably first order, that assert the existence of certain primitive objects with certain properties, and of certain primitive constructions on objects, such that all objects of interest can be constructed, ....

Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer, 1985. Grundlehren der mathematischen Wissenschafter, Volume 278.

.... [ F 0 S(C 0 ) F 0 H: Then one easily checks HomHA0 (H; K) HomHD0 (F 0 H;K) Finally, if C 2 CA 0 , then F 2 Omega QSD (C) QS(C) JC: Hence (2.10.1) 2.10.3) imply that F 2 Omega B QF 0 H = QH for general H. Xi This proof is obviously very general. Compare [2], x3.7, especially Theorem 2. x3. The equivalence of categories This section proves the statement that F 2 Omega B Q( Delta) HD L is an equivalence of categories. Recall that an object H is graded connected in HD if H 0 = F 2 in degree 0 and that HD HD is the full sub category of ....

M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der Mathematischen Wissenschaften 278, Springer-Verlag, Berlin 1985.

....algebraic theories by Freyd [14] theories formed with ECE equations [4] Jarzembski s weak varieties of partial algebras [22] and theories formed with strong equations [4, 20, 23, 31] Further, there are other logical frameworks capturing partiality as well. We only mention left exact sketches [3, 19], Coste s limit theories [9] and Meseguer s and Goguen s order sorted theories with sort constraints [18] It is a folklore theorem, that Hep theories are the same as left exact sketches, see [15, 32] And sort constraints of order sorted algebra are introduced to capture the same partiality ....

.... Delta Delta Delta e 0 m ) if for all valuations : X Gamma A satisfying all the e i (i = 1; n) there exists a unique extension : X[Y Gamma A of which satisfies all the e 0 i (i = 1; m) ut 3. 4 Left Exact Sketches The institution LESKET CH (left exact sketches, see [19, 3]) has signatures Sigma = G; U ) where G is a (finite) directed graph and U is a map assigning to each vertex (or object, to keep close to categorical terminology) a in G an arrow U (a) from a to a. Signature morphisms oe: G; U ) Gamma (G 0 ; U 0 ) are graph homomorphisms oe: G Gamma ....

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M. Barr, C. Wells. Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 278. Springer Verlag, 1985.

....there exists a sketch of Horn theories, which are known not to be equational in the sense of Lawvere (1963) The classical algebraic approach, as initiated by Lawvere (1963) is however a special case of the sketch approach. The standard reference on the basic theory of sketches is chapter 4 of (Barr and Wells 1985). Since we can view the operators in a signature as arrows in a graph, and limits are also eoeectively expressed as graphs the de nition of a sketch is the following. De nition 3.1.8 A LE sketch S is a 4 tuple S = hG; U; D; Ci, where G is a graph, U : N A is a function which takes each node n ....

Barr, M. and Wells, C.: 1985, Toposes, Triples and Theories, Vol. 278 of Grundlehren der mathematischen Wissenschaften, Springer Verlag, Berlin.

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M. Barr and C. Wells, Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 275, Springer-Verlag, Berlin (1985).

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M. Barr and C. Wells. Toposes, Triples and Theories. Grundlehren der mathematischen Wissenschaften 278. Springer Verlag, 1985.

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Barr, M. and Wells, C. Toposes, Triples, and Theories. Grundlehren der mathematischen Wissenschaften, vol. 278, Springer-Verlag, New York, 1985, xiii+345 pp.

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