## Metric, Topology and Multicategory - A Common Approach (2001)

Venue: | J. Pure Appl. Algebra |

Citations: | 12 - 7 self |

### BibTeX

@ARTICLE{Clementino01metric,topology,

author = {Maria Manuel Clementino and Walter Tholen},

title = {Metric, Topology and Multicategory - A Common Approach},

journal = {J. Pure Appl. Algebra},

year = {2001},

volume = {179},

pages = {13--47}

}

### Years of Citing Articles

### OpenURL

### Abstract

For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)-algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad in the bicategory of V-matrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of n-categories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.

### Citations

385 | Basic Concepts of Enriched Category Theory
- Kelly
- 1982
(Show Context)
Citation Context ...→ a(mX(X), x) (X ∈ T T X, x ∈ T X, x ∈ X), and with respect to these composition morphisms the commutativity conditions (10) and (11) simply become generalizations of the axioms for a V-catego=-=ry (see [15], [4]): αa ⊗ 1 a(x, x) ⊗ a(eX(x)-=-, x) ✲ T a(eT X(x), eX(x)) ⊗ a(eX(x), x) 1 ⊗ ux ✻ a(x, x) ⊗ I ∼ 11 ✲ ❄ a(x, x) c eT X(x),eX(x),x (9) (10) (11)sI ⊗ a(x, x) (T u)x ⊗ 1 T a((T eX)(x), x) ⊗ a(x, x) ✚ ✚✚✚✚... |

217 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ...e de Coimbra/FCT. † Partial financial assistance by NSERC is acknowledged. 1sa → b means a ≥ b, and in which the tensor product is given by addition), V-categories in the sense of Eilenberg and =-=Kelly [11] are nothing b-=-ut pairs (X, d) satisfying the basic “laws” 0 ≥ d(x, x), d(x, y) + d(y, x) ≥ d(x, z). For a general V-category A (with object set X), these are instances of the “operations” I → A(x, x) ... |

143 |
Introduction to bicategories, in
- Bénabou
(Show Context)
Citation Context ... discussion of the topics of this paper is in progress and must appear elsewhere. Hence, here we - do not discuss monads and related notions in the general context of bicategories or 2categories (see =-=[2]-=- and, for a recent account, [16]) but restrict ourselves to presenting them ad hoc as needed - present the 2-categorical structure of the category of reflexive, transitive (T, V)-algebras (=(T, V)-cat... |

114 |
Metric spaces, generalized logic, and closed categories
- Lawvere
- 1974
(Show Context)
Citation Context ... 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 m-=-akes 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 fund... |

71 |
Adjoint functors and triples
- Eilenberg, Moore
- 1965
(Show Context)
Citation Context ...ssets X which come with a tripart structure, given by a V-valued relation (=matrix, distributor, profunctor) interpreted as an “action” T X × X a −→ V, T X a −→+ X in the sense of Eilenbe=-=rg and Moore [12]-=-. The other two parts of this structure represent the two basic laws or operations encountered in all examples and are described by a generalized monad structure on a, where a is considered a 1-cell i... |

69 |
Handbook of Categorical Algebra 2: Categories and Structures
- Borceux
- 1994
(Show Context)
Citation Context ...(X), x) (X ∈ T T X, x ∈ T X, x ∈ X), and with respect to these composition morphisms the commutativity conditions (10) and (11) simply become generalizations of the axioms for a V-category (see =-=[15], [4]): αa ⊗ 1 a(x, x) ⊗ a(eX(x), x) ��-=-� T a(eT X(x), eX(x)) ⊗ a(eX(x), x) 1 ⊗ ux ✻ a(x, x) ⊗ I ∼ 11 ✲ ❄ a(x, x) c eT X(x),eX(x),x (9) (10) (11)sI ⊗ a(x, x) (T u)x ⊗ 1 T a((T eX)(x), x) ⊗ a(x, x) ✚ ✚✚✚✚❃ T 2... |

34 |
Variation through enrichment
- Betti, Carboni, et al.
(Show Context)
Citation Context ...phisms. The existence of an internal hom is used only to make sure that the tensorproduct commutes in each variable with colimits. The bicategory Mat(V) of V-matrices is defined in full generality in =-=[3]; h-=-ere we consider the more special case considered in [24] and take as its - objects sets, normally denoted by X, Y , · · ·, also considered as (small) discrete categories, and - arrows (=1-cells) r ... |

31 |
Relational algebras
- Barr
- 1970
(Show Context)
Citation Context ...cussed the similarity of the characterization of exponentiable morphisms in the categories of preordered sets, of topological spaces, and of all (small) categories. Generalizing Manes [22] and Barr’=-=s [1] w-=-ork for topological spaces, in [8] we succeeded to present Lowen’s approach spaces as lax algebras, already employing a general monad T rather than the ultrafilter monad, as suggested by George Jane... |

29 | Deductive systems and categories i - Lambek - 1968 |

21 |
Approach spaces: The missing link in the Topology-Uniformity-Metric triad
- Lowen
- 1997
(Show Context)
Citation Context ... 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 descr... |

20 |
T-catégories (catégories dans un triple), Cahiers de Topologie et Géométrie Différentielle XII(3
- Burroni
- 1971
(Show Context)
Citation Context ...pace” Set category multicategory ultracategory “T -category” V V-category V-multicategory V-ultracategory “(T, V)-category” Here M denotes the free-monoid monad on Set, which was used also b=-=y Burroni [7]-=-, Leinster [20] and Hermida [13] to describe multicategories. While their approach (working with the bicategory Span T (B) for a cartesian monad T on a category B with pullbacks) allows for a good def... |

19 |
On weak higher dimensional categories
- Hermida, Makkai, et al.
- 1997
(Show Context)
Citation Context ...ures 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 UX × X −→ 2, where UX is the set of ultrafilters on X satisfying the t... |

18 |
Topological features of lax algebras
- Clementino, Hofmann
(Show Context)
Citation Context ...erization of exponentiable morphisms in the categories of preordered sets, of topological spaces, and of all (small) categories. Generalizing Manes [22] and Barr’s [1] work for topological spaces, i=-=n [8] w-=-e succeeded to present Lowen’s approach spaces as lax algebras, already employing a general monad T rather than the ultrafilter monad, as suggested by George Janelidze in a seminar presentation at A... |

11 |
A triple theoretic construction of compact algebras
- Manes
- 1969
(Show Context)
Citation Context ...[26], [9] we discussed the similarity of the characterization of exponentiable morphisms in the categories of preordered sets, of topological spaces, and of all (small) categories. Generalizing Manes =-=[22] and-=- Barr’s [1] work for topological spaces, in [8] we succeeded to present Lowen’s approach spaces as lax algebras, already employing a general monad T rather than the ultrafilter monad, as suggested... |

10 | The convergence approach to exponentiable maps
- Clementino, Hofmann, et al.
(Show Context)
Citation Context ...s automatically providing notions like additive multicategory. Our main goal in this research, however, has from the beginning been the development of the notion of ultracategory. In our papers [26], =-=[9]-=- we discussed the similarity of the characterization of exponentiable morphisms in the categories of preordered sets, of topological spaces, and of all (small) categories. Generalizing Manes [22] and ... |

9 | General operads and multicategories
- Leinster
- 1998
(Show Context)
Citation Context ...gory multicategory ultracategory “T -category” V V-category V-multicategory V-ultracategory “(T, V)-category” Here M denotes the free-monoid monad on Set, which was used also by Burroni [7], L=-=einster [20]-=- and Hermida [13] to describe multicategories. While their approach (working with the bicategory Span T (B) for a cartesian monad T on a category B with pullbacks) allows for a good definition of inte... |

9 | Lax factorization algebras
- Rosicky, Tholen
(Show Context)
Citation Context ...nt examples to our list, such as (for V = Set) the “squaring monad” on Cat, with T X = X 2 and 2 = {· → ·}, which has recently been used to describe certain functorial weak factorization syste=-=ms (see [25]). I-=-n closing, in addition to Lawvere’s paper we wish to pay special tribute to Burroni’s 1971 paper [7] which we discovered only at the end of our work for this paper, but which touches upon many of ... |

8 |
Representable multicategories, Adv
- Hermida
(Show Context)
Citation Context ...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 (se=-=e [13], [14]). -=-Indeed, it is well known that a topological space may be completely described by a “convergence” relation, i.e., by a function UX × X −→ 2, where UX is the set of ultrafilters on X satisfying... |

4 |
Coproducts and ultrafilters
- Börger
- 1987
(Show Context)
Citation Context ...UX the (ultra)filter f(x) on Y , generated by {f(A) | A ∈ x}, i.e., B ∈ f(x) if and only if f −1 (B) ∈ x. Since U preserves finite coproducts, there is a uniquely determined monad structure on=-= U (see [5]). Explicitly, eX -=-: X → UX, mX : UUX → UX assign to x ∈ X the fixed ultrafilter eX(x) = • x , and to X ∈ UUX the filter sum mX(X) ∈ UX, with A ⊆ X lying in mX(X) precisely when lies in X. A ♯ = {x ∈ U... |

3 |
Convergence in exponentiable spaces, Theory Appl. Categories (electronic journal) 5
- Pisani
- 1999
(Show Context)
Citation Context ...= • x , and to X ∈ UUX the filter sum mX(X) ∈ UX, with A ⊆ X lying in mX(X) precisely when lies in X. A ♯ = {x ∈ UX | A ∈ x} It is known how to extend U to a functor U : Rel(Set) → Rel=-=(Set) (see [1], [23]): for r : X −→+ Y o-=-ne defines the relation Ur : UX −→+ UY by x(Ur)y :⇔ ∀B ∈ y : r o (B) ∈ x, with r o (B) = {x ∈ X | ∃y ∈ B : xry}. But since x is an ultrafilter, we always have r o (B) ∈ x or (X \ r... |

2 |
regular and dense generators, Cahiers Topologie Géom. Différentielle Catégoriques 32
- Börger, Tholen, et al.
- 1991
(Show Context)
Citation Context ...) given in Section 8 in case V is an atomic Boolean algebra provides guidance on how to extend U in case V = Set, PrSet, Cat, · · ·. For this we recall that an object c in V is connected (=“copri=-=me”, [6]) if V(c, -=-−) : V → Set preserves coproducts; that is, if every f : c → � ai in V factors uniquely through a uniquely determined coproduct injection. i∈I The category V is called based (see [6]) if eve... |

2 | Multicategories revisited, in: Contemporary Mathematics 92 - Lambek - 1989 |

2 |
Factorization systems and distributive laws, preprint
- Rosebrugh, Wood
(Show Context)
Citation Context ...ake sure that the tensorproduct commutes in each variable with colimits. The bicategory Mat(V) of V-matrices is defined in full generality in [3]; here we consider the more special case considered in =-=[24] and take a-=-s its - objects sets, normally denoted by X, Y , · · ·, also considered as (small) discrete categories, and - arrows (=1-cells) r : X −→+ Y are families of V-objects r(x, y) (x ∈ X, y ∈ Y )... |

1 |
The formal theory of monads II, preprint
- Lack, Street
- 2000
(Show Context)
Citation Context ...is paper is in progress and must appear elsewhere. Hence, here we - do not discuss monads and related notions in the general context of bicategories or 2categories (see [2] and, for a recent account, =-=[16]-=-) but restrict ourselves to presenting them ad hoc as needed - present the 2-categorical structure of the category of reflexive, transitive (T, V)-algebras (=(T, V)-categories) only briefly at the end... |

1 |
model categories, in: Abstracts of the International Conference on Category Theory
- Tholen, Injectives
- 2000
(Show Context)
Citation Context ...t, thus automatically providing notions like additive multicategory. Our main goal in this research, however, has from the beginning been the development of the notion of ultracategory. In our papers =-=[26]-=-, [9] we discussed the similarity of the characterization of exponentiable morphisms in the categories of preordered sets, of topological spaces, and of all (small) categories. Generalizing Manes [22]... |