Name
Abstract | ||
|---|---|---|
It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for ( , V)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical ( , V)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of ( , V)-categories. Lawvere in his 1973 paper Metric spaces, generalized logic, and closed categories formulates a notion of complete V-category and shows that for (generalised) metric spaces it means Cauchy completeness. This notion of completeness deserved the attention of the categorical community, and the notion of Cauchy-complete category, or Freyd-Karoubi complete category is well-known, mostly in the context of Ab-enriched categories. However, it never got the attention of the topo- logical community. In this paper we interpret Lawvere's completeness in topological settings. We extend Lawvere's notion of complete V-category to the (topological) setting of ( , V)-categories (for a symmetric and unital quantale V), and show that it encompasses well-known notions in topological categories, meaning weakly sober space in the category of topological spaces and con- tinuous maps, weakly sober approach space in the category of approach spaces and non-expansive maps, and Cauchy-completeness in the category of quasi-uniform spaces and uniformly contin- uous maps. We present also a first step towards a possible construction of completion. Indeed, in the setting of V-categories, it is well-known that the completion of a V-category may be built out of the Yoneda embedding X → VX op . In the ( , V)-setting, we could prove that V has a canonical ( , V)-categorical structure and that every ( , V)-category X has a canonical dual |
Year | DOI | Venue |
|---|---|---|
2009 | 10.1007/s10485-008-9152-5 | Applied Categorical Structures |
Keywords | Field | DocType |
\(\mathsf{V}\),-category,Bimodule,Monad,\((\mathbb{T},\mathsf{V})\),-category,Completeness,18A05,18D15,18D20,18B35,18C15,54E15,54E50 | Topology,Discrete mathematics,Enriched category,Embedding,Bimodule,Topological space,Cauchy distribution,Metric space,Completeness (statistics),Monad (functional programming),Mathematics | Journal |
Volume | Issue | ISSN |
17 | 2 | 0927-2852 |
Citations | PageRank | References |
5 | 0.73 | 8 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Maria Manuel Clementino | 1 | 61 | 25.61 |
| Dirk Hofmann | 2 | 73 | 25.09 |