Lawvere, F.W., Metric spaces, generalized logic, and closed categories, Rend. Semi. Mat. Fis. Milano, vol. 43, pp. 135-166, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... functors, 6 Changing V, 7 When V is a complete lattice, 8 Extending the ultrafilter monad when V is a lattice, 9 V categories and V multicategories, 10 Extending the ultrafilter monad when V is based, 11 V ultracategories, 12 2 cells in Alg(T , e, m;V) 1 Introduction In his famous 1973 article [19] Lawvere makes the point that categories should not be considered just as gadgets appearing in a third level of abstraction described by the sequence elements structures categories , but that fundamental structures are themselves categories . For his most eminent example, he lets the metric d ....

.... principle , in this paper we wish to show that Lawvere s categorical description of fundamental mathematical structures may be generalized quite dramatically, so as to include geometric structures like topological spaces and the much lesser known approach spaces (see [21] but also Lambek s [19] multicategories which enjoy renewed interest in higher dimensional category theory (see [13] 14] Indeed, it is well known that a topological space may be completely described by a convergence relation, i.e. by a function 2, where UX is the set of ultrafilters on X satisfying the two ....

F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166.

....the basic functors connecting them in general terms. This one additional ingredient is an arbitrary complete lattice V with a monoidal closed structure that takes the place of the 2 element chain which implicitly governs the axioms defining topological spaces. Lawvere in his fundamental paper [12] considered for V the extended real half line (ordered opposite to the natural order) and displayed individual metric spaces as V categories. In [7] we combined Barr s and Lawvere s ideas and introduced (T, V) algebras, for a monad T on Set and a symmetric monoidal closed category V, ....

....variable: b i = b i . 1.2 Examples. 1) Each frame ( complete lattice in which binary meets distribute over arbitrary joins) is symmetric monoidal closed, with# given by binary meet and k = the top element. In particular, the two element chain 2 = #, # carries this structure. 2) [12] Let be the extended real (half )line, provided with the order opposite to the natural order (so that a i = inf a i is the natural infimum of the elements a i ) and with# = 3 the addition (extended by a = a = and k = 0. In this way we consider as a symmetric monoidal closed ....

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F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166.

....is abstract nonsense taken to the max. The de ning characteristic of a V category or enriched category is that its morphisms from object a to object b band together not as a set but as an object of V , with the characteristic features of a category being reformulated in terms of V . Lawvere [14] has provided an attractive way of conceptually taming V categories, by viewing the objects of V as distances, and V categories as generalized metric spaces satisfying a suitable triangle inequality over such distances. Besides being good pedagogy, this view nicely connects enrichment with the ....

Lawvere, W.: Metric spaces, generalized logic, and closed categories. In: Rendiconti del Seminario Matematico e Fisico di Milano, XLIII. Tipogra a Fusi, Pavia (1973)

....obtain Cartesian closed categories. As the unifying concept of pre orders and metric spaces we follow Lawvere and use enriched categories. Enriched categories were introduced by Eilenberg and Kelly in 1966 ( Eilenberg Kelly 66] and popularized in the best sense of the word by Lawvere in 1973 ( Lawvere 73] In this latter paper Lawvere showed how to use essentially categorical tech niques on for instance pre orders and metric spaces by softening the requirements on what constitutes a category, such that the resulting concept, an enriched category includes pre orders, generalized) metric ....

....Allowing infinite distances the classifier of connections is then the extended interval [0, c] Rules for composition of connections amount to transitivity of pre orders and to the triangular inequality for metric spaces respectively. This approach to unification also suggests by itself ( Lawvere 73] p. 142) a suitable logic in which to reason about such structures, viz. intuitionistic logic in which the space of truth values is the classi fier of the connections. So we should treat pre orders in a two valued logic and (generalized) metric spaces in a [0, c] valued logic. In the cases where ....

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Lawvere, F.W., Metric spaces, generalized logic, and closed categories, Rend. Semi. Mat. Fis. Milano, vol. 43, pp. 135-166, 1974.

..... This question was largely answered in [KLSS99] with the introduction of the so called two sided enrichments. To explain partly these results, we should start from the definition of MonCat , the category of monoidal functors between monoidal categories [Ben63] and enrichments over them [EiKe66] [Law73]. A monoidal functor F : induces a 2 functor F : MonCat is equipped with a 2 categorical structure by defining 2 cells in it as monoidal natural transformations ( EiKe66] The process ( of sending to and F to F extends to a 2 functor from MonCat to 2 Cat . Adjunctions in MonCat ....

F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. di Milano 43 (1973), 135-166

....University road, Leicester LE1 7RH, England e mail: vs27 mcs.le.ac.uk 1 objects respectively of V Cat and W Cat. The rst point of our problem was answered in [KLSS99] Let us start from the de nition of monoidal functor between monoidal categories [Ben63] and enrichments over them [EiKe66] [Law73]. A monoidal functor F : V W induces a 2 functor F : V Cat W Cat. MonCat is equipped with a 2 categorical structure by de ning 2 cells in it as monoidal natural transformations ( EiKe66] The process ( of sending V to V Cat and F to F extends to a 2 functor from MonCat to 2 Cat. ....

F.W Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. di Milano 43, 73, 135-166

....step uses (I0) the third one uses (2 01) and the fourth one uses the naturality of u Pi . The other condition follows by symmetry. 3.05. Remarks. 0) Condition (b) of Proposition 3.04. directly translates the notion of bimodule familiar from algebra, cf. Examples 1.03. 2) 3.07. 1) as well as [7] and [3] We have dropped the prefix bi , because of it s different connotation in bicategory . The terms distributor and profunctor also appear in the literature. 1) Remark 3.01. 2) carries over to the monad setting. 3.06. Theorem. Monads in Y , m modules and modisms form a bicategory Y ....

....of Y mnd within Y int . 5. The inheritance of closedness The familiar notion of closedness for a symmetric monoidal category splits into two notions in the absence of symmetry: left closedness and right closedness. Their usefulness in connection with m modules was already observed by Lawvere [7]. These notions make perfect sense in any bicategory. Street and Walters [11] have formalized the corresponding generalizations in terms of the existence of all right extensions respectively all right liftings: 5.00. Definition. For 1 cells A r B and A t C , a right extension along r of ....

Lawvere, F. W. Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 (1973), 135--166.

....step uses (I0) the third one uses (2 01) and the fourth one uses the naturality of u Pi . The other condition follows by symmetry. 3.05 Remarks. 0) Condition (b) of Proposition 3.04 directly translates the notion of bimodule familiar from algebra, cf. Examples 1.03(2) 3. 07(1) as well as [7] and [3] We have dropped the prefix bi , because of it s different connotation in bicategory . The terms distributor and profunctor also appear in the literature. 1) Remark 3.01(2) carries over to the monad setting. 3.06 Theorem. Monads in Y , m modules and modisms form a bicategory mnd ....

....of mnd (Y) within int (Y) 5 The inheritance of closedness The familiar notion of closedness for a symmetric monoidal category splits into two notions in the absence of symmetry: left closedness and right closedness. Their usefulness in connection with m modules was already observed by Lawvere [7]. These notions make perfect sense in any bicategory. Street and Walters [11] have formalized the corresponding generalizations in terms of the existence of all right extensions respectively all right liftings: 22 5.00 Definition. For 1 cells A r B and A t C , a right extension along r of t ....

Lawvere, F. W. Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 (1973), 135--166.

....obtain Cartesian closed categories. As the unifying concept of pre orders and metric spaces we follow Lawvere and use enriched categories. Enriched categories were introduced by Eilenberg and Kelly in 1966 ( Eilenberg Kelly 66] and popularized in the best sense of the word by Lawvere in 1973 ( Lawvere 73] In this latter paper Lawvere showed how to use essentially categorical techniques on for instance pre orders and metric spaces by softening the requirements on what constitutes a category, such that the resulting concept, an enriched category includes preorders, generalized) metric spaces, ....

....Allowing infinite distances the classifier of connections is then the extended interval [0; 1] Rules for composition of connections amount to transitivity of pre orders and to the triangular inequality for metric spaces respectively. This approach to unification also suggests by itself ( Lawvere 73] p. 142) a suitable logic in which to reason about such structures, viz. intuitionistic logic in which the space of truth values is the classifier of the connections. So we should treat pre orders in a two valued logic and (generalized) metric spaces in a [0; 1] valued logic. In the cases where ....

[Article contains additional citation context not shown here]

Lawvere, F.W., Metric spaces, generalized logic, and closed categories, Rend. Semi. Mat. Fis. Milano, vol. 43, pp. 135-166, 1974.

....a of X is the supremum of the character iff a is an upper bound of and for all x 2 X, it is true that sup y2X i d(y; x) Gamma (y) j d(a; x) 3 Assume X and Y are V continuity spaces, f : X Y is a nonexpansive map and is a character on X. Then the direct image of under f [Law73], denoted by b f( is the function from Y to V defined by b f ( y) inf x2X i (x) d(y; f(x) j ; for y 2 Y : It is easy to show that b f( is a character on Y , that the resulting map b f : c X b Y is nonexpansive, that the assignment f 7 b f is functorial, and that ....

F.W. Lawvere. Metric spaces, generalized logic, and closed categories. Rend. Semi. Mat. Fis. Milano, 43:135--166, 1973.

....bed. Linear logic, for instance, does not arise entirely in terms of adjunctions. But for the basic constructs, the adjunctions seem to be working remarkably well. In [32] Lawvere has described the comprehension scheme, assigning to each predicate P (x) a set fxjP (x)g, as an adjoint functor. In [33], he has explained how maps, as total and singlevalued relations, can be presented as self adjoint bimodules. This idea is central in the present paper. In the next three sections, we introduce the categorical setting for generalized regular logic, present some typical examples, and describe the ....

....in C the cartesian product, and provides the fibrewise products in . The direct image of a 1 cell P 2 (A) along a map R 2 R(A;B) will be the 1 cell PR 2 (B) The inverse image of Q 2 (B) will be the 1 cell QR 2 (A) where R is the right adjoint of R, which makes it into a map. Maps. In [33], Lawvere has defined a map as a 1 cell R : A j B, such that R o : B j A is its right adjoint. This just means that there are proofs (13) and (14) satisfying (jR ; R ) id R : 17) The other adjunction equation (R o j; R o ) id R o follows by dualizing. Actually, one could start ....

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F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43(1973), 135--166

....of chains of coreflections (adjunctions) can be calculated as follows. Let Fn : A n Fnan Fnan ## Fnan Fnan cc A n 1 : Gn be an chain in Prof cor (Prof adj ) For every n, write En : A n , A n for the embedding of the category A n into its Cauchy completion A n [18], and let E n : c A n Gamma c A n denote the induced equivalence of categories. It is known (see, for example, 18, 3] that there exist functors Hn : A n A n 1 such that Fn can be seen as a left Kan extension [21, 3, 16] as follows: Fn = Lan yn (E n 1 HnEn ) Let n ....

....A n 1 : Gn be an chain in Prof cor (Prof adj ) For every n, write En : A n , A n for the embedding of the category A n into its Cauchy completion A n [18] and let E n : c A n Gamma c A n denote the induced equivalence of categories. It is known (see, for example, [18, 3]) that there exist functors Hn : A n A n 1 such that Fn can be seen as a left Kan extension [21, 3, 16] as follows: Fn = Lan yn (E n 1 HnEn ) Let n : A n A be a colimit of the chain hHn : A n A n 1 i in Cat. Then, a pseudo colimit in Prof of the chain Fn : A n ....

[Article contains additional citation context not shown here]

F. W. Lawvere. Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano, 43:135--166, 1973.

....bed. Linear logic, for instance, does not arise entirely in terms of adjunctions. But for the basic constructs, the adjunctions seem to be working remarkably well. In [32] Lawvere has described the comprehension scheme, assigning to each predicate P (x) a set fxjP (x)g, as an adjoint functor. In [33], he has explained how maps, as total and single valued relations, can be presented as self adjoint bimodules. This idea is central in the present paper. In the next three sections, we introduce the categorical setting for generalized regular logic, present some typical examples, and describe the ....

....in C the cartesian product, and provides the fibrewise products in . The direct image of a 1 cell P 2 (A) along a map R 2 R(A;B) will be the 1 cell PR 2 (B) The inverse image of Q 2 (B) will be the 1 cell QR 2 (A) where R is the right adjoint of R, which makes it into a map. Maps. In [33], Lawvere has defined a map as a 1 cell R : A j B, such that R o : B j A is its right adjoint. This just means that there are proofs (4.3) and (4.4) satisfying (jR ; R ) id R : 4.7) The other adjunction equation (R o j; R o ) id R o follows by dualizing. Actually, one could start ....

[Article contains additional citation context not shown here]

F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43(1973), 135--166

....of the monoid. Although Eilenberg and Kelly described enriched categories in their full generality, their motivating applications were confined to V s forming classes rather than sets, an emphasis continued in Kelly s subsequent book on enriched categories [Kel82, p22] Yet in 1974 F.W. Lawvere [Law73] in an excellent advertisement for the utility of enriched categories emphasized V s that were semirings, hence sets rather than classes, and with category structure merely that of a poset. This brought enriched categories into close proximity to the parallel computer science development. ....

....monoidal structure being 1 ( Gamma1) 1, this being the one bit of asymmetry in R. As we have seen, R op is the monoidal structure associated with ordinary metric spaces. A sometimes useful closed subcategory of R op , namely R op , is obtained by omitting the negative reals and Gamma1 [Law73]. The internal homfunctor then becomes truncated subtraction, in which negative differences are rounded up to 0. This is the category of generalized metric spaces; if the further restrictions are made that distances be symmetric and distinct points are a nonzero distance apart then we have the ....

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W. Lawvere. Metric spaces, generalized logic, and closed categories. In Rendiconti del Seminario Matematico e Fisico di Milano, XLIII. Tipografia Fusi, Pavia, 1973.

....can elegantly and usefully do both at once, giving rise to enriched categories. The basic ideas behind enriched categories can be traced to Mac Lane [Mac65] with much of the detail worked out by Eilenberg and Kelly [EK65] with the many subsequent developments condensed by Kelly [Kel82] Lawvere [Law73] provides a highly readable account of the concepts. We require of the edge labels only that they form a monoidal category. Roughly speaking this is a set bearing the structure of both a category and a monoid. Formally a monoidal category D = hD; Omega ; I ; ff; aei is a category D = hD 0 ; ....

.... more accessible and appealing is the very pretty case D = R op 0 = hhR 0 ; i; 0i, reverse ordered nonnegative reals under addition, for which R Cat becomes the category of (generalized) metric spaces, with the composition law as the triangle inequality and functors as contracting maps [Law73]. Enriched categories first appeared in computer science with D = Poset = hPoset ; Theta; 1i [Wan79] yielding order enriched categories, a natural notion for domain theory. Poset itself is definable as (the antisymmetric subcategory of) hhf0; 1g; i; 1i Cat, categories enriched in truthvalues. ....

W. Lawvere. Metric spaces, generalized logic, and closed categories. In Rendiconti del Seminario Matematico e Fisico di Milano, XLIII. Tipografia Fusi, Pavia, 1973.

....of abelian groups, and models of linear logic. Hence the paper is written using the weaker notion of tensor product (briefly introduced below) If monoidal categories are unfamiliar then this section may be skipped and the rest of the paper read by interpreting Omega as Theta and I as 1. Recall [26, 19] that a symmetric monoidal category (V; Omega ; I; a; l; r; c) is given by a category V equipped with a binary functor Omega : V 2 V called the tensor product which is associative, unitary (with unit object I) and symmetric, up to natural isomorphisms whose components are given by a A;B;C ....

F.W. Lawvere, Metric Spaces, Generalized Logic, and Closed Categories, Rend. del Sem. Mat. e Fis. di Milano 43 (1973) 135--166.

....will introduce you to monoidal categories, and Chapter X to Kan extensions. The new edition [Mac97] also contains important material on topos theory, 2categories, bicategories, and presheaves. Beyond that, I would recommend categorical logic and fibrations [Jac99, Pho92] enriched category theory [Law73a], and any further writing by Lawvere such as [Law70, Law69, Law91] Links . http: www.mta.ca #cat dist categories.html . http: www.acsu.buffalo.edu #wlawvere ....

F. William Lawvere. Metric spaces, generalized logic, and closed categories. In Rendiconti del Seminario Matematico e Fisico di Milano, XLIII. Tipografia Fusi, Pavia, 1973.

....nor is continuity explicity proved. However, this is justified by adherence to a constructivist discipline that makes topology implicit. Technically, we use an approach to completion that derives from enriched category theory this was first proposed in the study of quasimetric spaces in [11]. According to our definition, the points of the completion are the flat left modules over the quasimetric space, in a sense that is already well known for categories enriched over Abelian groups or sets. However, the clearest topological way of understanding this is that instead of constructing ....

....However, an equivalent topological view is that M describes the open balls B e (x) that contain the point (i.e. for which M(x) e) and our definition will fairly naturally turn out to be equivalent to filters of such open balls (Proposition 4. 8) Quasimetric spaces via enriched categories In [11] metric spaces are discussed as V enriched categories (V here being the extended non negative real line [0, and despite the abstractness of this account it has a solid conceptual basis: it views a metric space as a set in which equality formulae receive their truth values as real numbers. The ....

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F.W. Lawvere, Metric spaces, generalised logic, and closed categories, Rend. del Sem. Mat. e Fis. di Milano 43 (1973).

....enriched categories to quasi uniform spaces. Vincent Schmitt, Department of mathematics and computer science, University of Leicester, University road, LEICESTER LE1 7RH, Email: vs27 mcs.le. ac.uk 03 07 00 Abstract: The represention of complete metric spaces of [Law73] by enrichments is extented to quasi uniform spaces. Moreover quasi uniformly continuous maps are described as enriched functors. The quasi uniform space completion is also viewed as a Cauchy completion. Super monoidal functors are introduced to obtain these results. A 2 category of enrichments ....

....introduced to obtain these results. A 2 category of enrichments over di erent bases is de ned. In this general context the Cauchy completion is still a universal construction. keywords: Enriched categories, change of base, Cauchy completion, uniform and quasi uniform spaces. 1 Introduction In [Law73] the Cauchy completion for enrichments over monoidal closed categories is de ned. This completion process is shown to capture the metric space completion: generalised metric spaces correspond to enrichments and the complete metric spaces correspond to Cauchy complete enrichments. In this paper, ....

[Article contains additional citation context not shown here]

F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. di Milano 43, 73, 135-166

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Lawvere, F.W., Metric spaces, generalized logic, and closed categories, Rend. Semi. Mat. Fis. Milano, vol. 43, pp. 135-166, 1974.

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F.W. Lawvere, Metric spaces, generalised logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135--166.

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F.W. Lawvere, Metric Spaces, Generalized Logic, and Closed Categories, Rend. del Sem. Mat. e Fis. di Milano 43 (1973) 135--166.

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F. W. Lawvere. Metric spaces, generalized logic, and closed categories. In Rend. del Sem. Mat. e Fis. di Milano, volume 43, pages 135--166, 1973.

No context found.

F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166.

No context found.

F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166.

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