Lawvere, F.W. (1963), Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. 50, 869--872.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the interleaving parallel connection, which we might denote Y , as opposed to the true concurrent connection that we have denoted X k Y and for which we have given a universal characterision above. It is interesting to note that forming the so called Lawvere algebraic theory (in the sense of [28] and [29] of a hidden sorted speci cation gives rise to a structure generalising some traditional structures of process algebra, in that there is an action of the monoid of all method expressions (in the sense of FOOPS [21] on the collection of all attribute expressions (again in the sense of ....

F. William Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, U.S.A., 50:869-872, 1963. Summary of Ph.D. Thesis, Columbia University.

....authors to use monads, even if then only named standard constructions . Huber in 1961 showed that adjoint pairs give rise to monads. Kleisli [18] and Eilenberg and Moore [7] proved the converse in 1965. The construct of a Kleisli category was thus made explicit in those contributions. Lawvere [19] introduced universal algebra into category theory. This can be seen as the birth of the term monad. These developments then contain all categorical elements for substitution theories. The exploitation of terms and unifications within logic programming is formally described in [22] as early as in ....

F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963.

....nite products. Moreover, we derive a categorical equivalence of a particular subcategory of algebras and of these presheaves, which gives an alternative coalgebraic presentation of algebras (this equivalence is a slight generalization of the Lawvere s characterization of many sorted algebras [12]) At a more intuitive level, this gives another evidence that the algebraic observational equivalence is of essentially coalgebraic nature. We conclude the paper with construction of an adjunction between a cocomplete model category and a corresponding category of presheaves, where right ....

....(Lemma 6.2) that each presheaf preserving nite products corresponds to some algebra. In particular, note that each such presheaf is automatically rooted, i.e. maps initial algebra (empty coproduct) in to a singleton. This fact is closely related to the standard Lawvere s representation ([12]) of models of algebraic theories as nite product preserving functors. Lemma 6.2 A presheaf over corresponds to a algebra i it preserves nite products. One way of proving this is by exploiting the Lawvere s characterization. Alternatively, a direct proof can be based on the following ....

Lawvere, F., W. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. USA 52, pp. 869-873, 1963.

....the internal language) and the two processes are mutually inverse. Part of the appeal of this concept is that models and translations become essentially the same notion (namely, structure preserving functors) These ideas originated with Lawvere s discovery of the so called algebraic theories [14], later to be extended to an equivalence between categories with products and multi sorted algebra [15] There is, by now, a whole list of results of this type, including the equivalence of typed calculus and cartesian closed categories [13] and categorical representations of more complex type ....

F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Natl. Acad. Sci. USA 50 (1963), 869-872.

....the terms of the calculus, so the natural categorical constructions on these ordered categories [33,39,48,47,75,72] implicitly contain rewrite relations whose equational theory matches that suggested by the traditional semantics. Although these structures, and the more general enriched categories [24,50,51, 56], are beginning to gain wider acceptance, their use is still limited enough to warrant formal definitions. These are given here, although a discussion of the use of ordered categories in modelling term rewriting in the calculus is postponed until the next chapter. Definition 1.1.1 An ordered ....

....rewrite relations and so the conceptually simpler ordered categories are preferred. Having said this, our arguments can of course be generalised to the 2 categorical setting. Another possibility for modelling term rewriting systems is to generalise the traditional approach of functorial semantics [6,34,56] by considering a model of a term rewriting system as being a structure preserving functor from a free theory to a category with appropriate structure. This approach, although well suited to prov ing results about the category of models of a rewriting system, does not provide any intuition as to ....

F. W. Lawvere. Functorial semantics of algebraic theories. In Proceedings of the National Academy of Sciences, 50:869--872, 1963.

....of atomic symbols. The soundness assures the validity of simulation methods to data refinement. Later, Single completeness theorem rather than Joint completeness theorem is argued in [2,13] The aim of this paper is to give a foundation of the functional semantics approach, as in [5,7], to the calculus, which we call calculus, introduced in [3] Functorial semantics given by adjunction Interpretations of both atomic symbols and commands are given by functors. An existence of adjunction enables the extension of an interpretation of atomic symbols to that of commands. ....

W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad.Sci. U.S.A., 50 (1963) 869--873.

....that, given any function f : X Y (in SET) the following diagram commutes, X Theta X f Theta f Y Theta Y aX a Y f X Y where X ;Y are lists of X;Y , and where aX ; a Y are their append operations. It is easy to prove this fact in general, making use of Lawvere s view [40] that an algebraic theory is a category with operations as its morphisms, an algebra is a product preserving functor (to SET) and a homomorphism is a natural transformation. Lawvere s inisght that homomorphisms are natural transformations is at first surprising, but is actually fairly simple and ....

F. William Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, U.S.A., 50:869--872, 1963. Summary of Ph.D. Thesis, Columbia University.

....Note that we use diagrammatic order for the horizontal , ff fi, and vertical , fl; ffi, composition of natural transformations [102] 2.4 2 Category Models This section can be skipped in a first reading; it provides useful background for the discussion on tile models in Section 3.13. Lawvere [89] made the seminal discovery that, given an equational theory T = Sigma ; E) and a Sigma algebra A satisfying E, the assignment to each E equivalence class [t(x 1 ; xn ) of its associated functional interpretation in A, A [t] A n Gamma A is in fact a product preserving functor A ....

F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869--873, 1963. Summary of Ph.D. Thesis, Columbia University.

....A to their (non annotated) counterparts of Rth. In other words, erases the control information, leaving the logical information untouched. More precisely, acts as a special case of mapping from lists of terms to lists of terms (in the Lawvere theories associated to the equational signature [16]) such that singleton lists of terms are mapped either to singleton lists of terms or to the empty sequence, whereas a non singleton list, say [e 1 ; e n ] is mapped to the concatenation of the images of the singleton lists of the components, i.e. e 1 ] e n ] By ....

F. William Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869-873, 1963. Summary of Ph.D. Thesis, Columbia University.

....are various kinds of monads on C. Example 1.8 (Theories) Monoids in the category of finitary endofunctors are finitary monads. In the case of finitary endofunctors on Ens the category of finitary monads is equivalent to the category of finitary algebraic theories in the sense of Lawvere [20]. In this particular case, coefficients turn out to be the general coefficients for cohomology of algebraic theories briefly mentioned in [14] Example 1.9 (Operads) In example 1.7, let C be the category of vector spaces over a characteristic zero field k. Consider the full subcategory of End(C) ....

F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA 50 (1963), 869--872.

....of the commutator calculus, i.e. properties which do not depend on a particular choice of basic operations for the algebras of the variety. A good example of such approach is [Pe] From this viewpoint, it is natural to represent varieties by the gadgets called finitary algebraic theories (see [L]) these are essentially just categories conisiting of finite cartesian powers of a single object. Considering theories leads to another natural definition for nilpotence of varieties, using the notion of linear extension of a category, in the sense of [BW] Such definition is inherent in [JP] ....

F. W. Lawvere, Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. 50 (1963) 869--872.

.... coecients in a bifunctor Hom R (I; T ) classi es linear extensions of the category M R by the bifunctor: Hom R (I; T ) E M R : We prove that here E will always be equivalent to the category of all nitely generated free models of some uniquely determined algebraic theory in Lawvere s sense [12,21,25]. The functor Rings Theories from the category of associative rings with unit to the category of algebraic theories, which assigns the theory of left R modules to the ring R, is known to be a full embedding. This enables us to identify rings with corresponding theories. Hence it turns out that ....

F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA 50 (1963), 869-872.

....N# # PN and set M## # N M# # N#. The derived smash product is well de ned up to stable equivalence. There are certain standard Tor spectral sequences converging to the homotopy groups of M## # N, see [37, Lemma 3.1] 2. Algebraic theories Algebraic theories, introduced by Lawvere [23], formalize the concept of an algebraic object as a set together with n ary operations for various n3# and equational relations. A detailed exposition of algebraic theories can be found in [6, Section 3] To do homotopy theory, we use algebraic theories which are enriched over the category of ....

F.W. Lawvere, Functorial semantics of algebraic theories, Proceedings of the National Academy of Sciences U.S.A. 50 (1963) 869}872.

....model (defined formally below) can be viewed as an extension of the widely used EntityRelationship (ER) approach [5] to information modelling. In addition to the graphical advantages of the ER approach, the SDM incorporates many of the insights of categorical universal algebra [3] chapter 4) [24], allowing a much more detailed specification of constraints, and a commensurate improvement in the quality of the specification and design process. Thus, the principal successes of the SDM approach have been in the early design stage. If the design is intended to be implemented, the model is ....

F.W. Lawvere. Functorial semantics of algebraic theories. PhD thesis, Columbia University, 1963.

....and Horn clauses; Section 1.2 recalls the operational machinery of logic programming; Section 1.3 presents the tile notation and the categorical models based on double categories; Section 1.4 presents the concepts of Section 1. 1 under a different light (exploiting Lawvere s pioneering work (Lawvere, 1963)) which will offer a more convenient notation for representing logic programs in tile logic. While the contents of Sections 1.1 and 1.2 are standard, the notions recalled in Sections 1.3 and 1.4 might be not so familiar to the logic programming community. 8 R. Bruni, U. Montanari and F. Rossi ....

....decomposition is enough for guaranteeing the congruence of tile bisimilarity and tile trace equivalence w.r.t. these closure operations. 1.4 Algebraic theories An alternative presentation of the category of substitutions discussed in Section 1. 1 can be obtained resorting to algebraic theories (Lawvere, 1963). Interactive Semantics 15 op f 2 Sn f : n 1 id n 2 N id n : n n seq a:n m b:m k a;b:n k mon a:n m b:k l a b:n k m l sym n;m 2 N g n ;m : n m m n dup n 2 N n : n n n dis n 2 N n : n 0 Figure 7. The inference rules for the generation of Th[S] Remark ....

Lawvere, F.W. (1963). Functorial semantics of algebraic theories. Proc. national academy of science, 50, 869--872.

....given an algebraic theory T, then the category Mod(T) of models, i.e. set valued functors on T preserving nite products, is a variety. And every variety has an algebraic theory, i.e. is equivalent to Mod(T) for some T. This has been shown in the by now classical dissertation of F. W. Lawvere [L]. However, a variety has typically many (non equivalent) algebraic theories. In [GU] it has been proved that all algebraic theories of a given variety V have the same Cauchy completion (where a category is Cauchy complete if it has split idempotents, and a Cauchy completion of a category is a re ....

Lawvere F. W., Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963

....unary context; it can be equivalently written f(x 1 ; f(x 1 ; x 1 ) Substitutions and their composition ; form a (cartesian) category Subs , with linear substitutions forming a monoidal subcategory. An alternative presentation of Subs can be obtained resorting to Lawvere s algebraic theories [29]. The free algebraic theory associated to a signature defines a cartesian category denoted by Th[ its objects are underlined natural numbers; its arrows from m to n are in a one to one correspondence with n tuples of terms of the free algebra with (at most) m variables, and composition of ....

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869--872, 1963.

....they can compose, decompose and even collapse in many uncontrolled and overlapping ways. However, we shall put such complex combinatorics under the control framework provided by the categorical approach to equational logic: such approach goes back to the classical pioneering paper of F.W. Lawvere [9] in functorial semantics. 2 Basically, equational theories are identi ed with categories with products, so that in our situation we need to manipulate presentations of pushouts among such categories. We get a rst general and simple presentation of these pushouts in Section 3 by means of ....

F. William Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50:869-872, 1963.

....mathematical background of our work. In Section 4, we define models (semantics) of our example language in terms of Fix algebras, which are given by algebraic structure on locally ordered categories. The technique we use here is an enriched version of functorial semantics as originated by Lawvere [10] (see also [11] In Section 5, we define refinement as a structure respecting lax transformation between models and prove soundness and completeness of lax transformations with respect to refinements. The fundamental mathematical fact we need here is that the adjunction between the category ....

F. William Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50:869--873, 1963. 14

....to a category in the sense of category theory, for example the numeric data elds. The order in which the numeric elds occur and the interpretation (convention or policy) determining which is the month and which is the day are functors. The data (numbers) are in category theory objects. Lawvere [19] has introduced the concept of a natural number object where the implicit ordering of the integers are ordinary categorial arrows. From the IRDS perspective, a date is some measure of time at the conceptual level of IRDDS. The constructs available at the IRDD level are giga years, years since ....

Lawvere, B, Functorial Semantics of Algebraic Theories, Proc. National Academy of Sciences USA, 50(1) 869-872 (1963).

.... together with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [37,70] to the more recent [41,48] Then, our definition of algebraic theory can be proved equivalent to the classical one, dating back to the early work of Lawvere [47,50]. The following, classical result states the equivalence between these theories and the usual term algebra construction for ordinary signatures. Proposition 2.1 (algebraic theories and term algebras) Given an ordinary signature Sigma, for all n; m 2 N c there exists a one to one ....

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869--872, 1963.

....such definitions are slightly more involved than the ones above, they have the advantage of separating in a better way the Sigma structure from the additional algebraic structure that the carrier can enjoy. We start defining the Lawvere theory Th( Sigma) associated with a signature Sigma [12]. This is a cartesian category having natural numbers as objects, generated in a free way from the operators of Sigma . The relevant property (on which we will come back later) is that arrows from m to 1 are in one to one correspondence with terms of the free Sigma algebra with at most m ....

....hold. Moreover, let S CPO Mod Sigma be the subcategory of CPO Mod Sigma having the same objects and as arrows natural transformations such that all components are strict continuous functions. Then Sigma SCAlg = S CPO Mod Sigma . ut The result for Set models was proved by WilliamLawvere [12] in his doctoral dissertation, and is well known: here we only sketch the underlying ideas. Given a Set model M: Th( Sigma) Set, consider the pair AM = hM(1) ae = fM(f Sigma ) j f 2 Sigma gi. It is easy to check that AM is a Sigma algebra: indeed, M(1) is a set, and for any f 2 Sigma n , ....

F. W. Lawvere, Functorial Semantics of Algebraic Theories, in Proc. National Academy of Science 50, 1963, pp. 869--872.

....t i is linear and var(t i ) var (t j ) for i 6= j. Substitutions and their composition ; form a (cartesian) category Subs , with linear substitutions forming a monoidal subcategory. An alternative presentation of Subs can be obtained resorting to algebraic theories. An algebraic theory [15] is a cartesian category having underlined natural numbers as objects. The free algebraic theory associated to a signature is denoted by Th[ the arrows from m to n are in a one to one correspondence with n tuples of terms of the free algebra with (at most) m variables, and composition of ....

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869-872, 1963.

....interpretations of atomic symbols. The soundness assures the validity of simulation methods to data re nement. Later, Single completeness theorem rather than Joint completeness theorem is argued in [2,4] The aim of this paper is to give a foundation of the functional semantics approach, as in [5,7], to the calculus, which we call calculus, introduced in [3] Functorial semantics given by adjunction Interpretations of both atomic symbols and commands are given by functors. An existence of adjunction enables the extension of an interpretation of atomic symbols to that of commands. ....

W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad.Sci. U.S.A., 50 (1963) 869-873.

.... in Giraud s theorem because generators are no longer indecomposable) On the algebraic side of the world, we have Lawvere s theorem which states that a category is equivalent to an algebraic one if and only if it is exact, and has a finitely presentable regular projective regular generator [6]. Lawvere s theorem is the non additive extension of Gabriel Mitchell s characterization of module categories. Localizations of module categories are exactly Grothendieck categories (Gabriel Popescu [3] and the nonadditive version of Gabriel Popescu s theorem states that exact categories with a ....

F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. 50 (1963) 869-872.

....to the characterization of rational terms, focussed instead on the extension of the notion of signature by means of suitable recursion operators, and on an axiomatic characterization of unique fixed points. A seminal stream (with tight links to the categorical notion of algebraic theories [24]) started with the paper on algebraic iterative theories by Elgot [13] Here we recall just a few basic results, for which we refer the reader to [4] Definition 4 ( terms) Let Sigma be a signature and X be a (countably infinite) set of variables such that Sigma X = The set T Sigma (X) ....

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869--872, 1963.

....set of cells is in one to one correspondence with the rewrite rules in R, and, moreover, the underlying category is able to describe in a faithful way the structure of the word algebra associated to a given signature. Given a signature Sigma , we denote by Th( Sigma) the associated Lawvere theory [Law63]: it is a category with finite products having natural numbers as objects, freely generated from the operators of Sigma . Definition9 (Lawvere Theories) Given a signature Sigma , the associated Lawvere theory is the category Th( Sigma) with finite products such that 10 its objects are ....

F. W. Lawvere, Functorial Semantics of Algebraic Theories, Proc. National Academy of Science 50, 1963, pp. 869-872.

....usually are more descriptive than their algebraic counterparts. 7 b a f b C C C C b = a f b a F F F F x x x x a f b a f b = a Fig. 6. The naturality axiom for duplicators and dischargers. 2. 1 Enriching the Monoidal Structure The constructive de nition of algebraic theories [24] as symmetric monoidal categories, enriched with two natural transformations (in the terminology we adopt in the following, the duplicator and the discharger ) dates back to the mid Seventies [21,33] even if it has received a new stream of attention in these days. With respect to the usual ....

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869-872, 1963.

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Lawvere, F.W. (1963), Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. 50, 869--872.

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F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869--873, 1963.

No context found.

Lawvere, F. W. 1963. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences 50, 869--873.

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F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869--873, 1963.

No context found.

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869-872, 1963.

No context found.

F. Lawvere, Functorial semantics of algebraic theories, Proc. National Academy of Sciences 50 (1963) 869--872.

No context found.

F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869-873, 1963.

No context found.

F. William Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50:869--872, 1963.

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F. W. Lawvere. Functorial Semantics of Algebraic Theories. In Proc. Nat. Acad. Sci. U. S. A., volume 50, pages 869-872, 1963.

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F. Lawvere, Functorial semantics of algebraic theories, Proc. National Academy of Sciences 50 (1963) 869--872.

No context found.

F.W. Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50:869-873, 1963.

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F. W. Lawvere. Functorial semantics of algebraic theories. Proc. Natl. Acad. Sci. U.S.A., (50):869-872, 1963.

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F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869--873, 1963.

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F.W. Lawvere. Functorial Semantics of Algebraic Theories. Proc. National Academy of Science 50, 1963, pp.869-872.

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F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869-872.

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Lawvere, F.W. (1963), Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. 50, pp869--872.

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F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869-873, 1963.

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F.W. Lawvere. Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50:869--873, 1963.

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F. W. Lawvere. Functorial semantics of algebraic theories. Proceedings, National Academy of Sciences, 50:869--873, 1963.

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F.W. Lawvere. Functorial semantics of algebraic theories. Proceedings of the National Academy of Sciences, U.S.A., 50:869--872, 1963.

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F. W. Lawvere. Functorial semantics of algebraic theories. In Proc. National Academy of Science, U.S.A., 50, pages 869-872. Columbia University, 1963.

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F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University 1963.

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