For any suitable base category $mathcal{V} $, we find that $mathcal{V} $-fully faithful lax epimorphisms in $mathcal{V} $-$mathsf{Cat} $ are precisely those $mathcal{V}$-functors $F colon mathcal{A} to mathcal{B}$ whose induced $mathcal{V} $-functors $mathsf{Cauchy} F colon mathsf{Cauchy} mathcal{A} to mathsf{Cauchy} mathcal{B} $ between the Cauchy completions are equivalences. For the case $mathcal{V} = mathsf{Set} $, this is equivalent to requiring that the induced functor $mathsf{CAT} left( F,mathsf{Cat}right) $ between the categories of split (op)fibrations is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations $mathcal{F}$ to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) $mathcal{F}$-descent morphisms in the category of small categories $mathcal{Cat}$; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations -- previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.
{"title":"Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations","authors":"Fernando Lucatelli Nunes, Rui Prezado, L. Sousa","doi":"10.36045/j.bbms.221021","DOIUrl":"https://doi.org/10.36045/j.bbms.221021","url":null,"abstract":"For any suitable base category $mathcal{V} $, we find that $mathcal{V} $-fully faithful lax epimorphisms in $mathcal{V} $-$mathsf{Cat} $ are precisely those $mathcal{V}$-functors $F colon mathcal{A} to mathcal{B}$ whose induced $mathcal{V} $-functors $mathsf{Cauchy} F colon mathsf{Cauchy} mathcal{A} to mathsf{Cauchy} mathcal{B} $ between the Cauchy completions are equivalences. For the case $mathcal{V} = mathsf{Set} $, this is equivalent to requiring that the induced functor $mathsf{CAT} left( F,mathsf{Cat}right) $ between the categories of split (op)fibrations is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations $mathcal{F}$ to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) $mathcal{F}$-descent morphisms in the category of small categories $mathcal{Cat}$; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations -- previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76412035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension 4 p for a prime p > 5. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.
{"title":"Classification of finite-dimensional Hopf algebras over dual Radford algebras","authors":"Rongchuan Xiong, N. Hu","doi":"10.36045/j.bbms.210612","DOIUrl":"https://doi.org/10.36045/j.bbms.210612","url":null,"abstract":". We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension 4 p for a prime p > 5. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"21 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73951727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cofiniteness of generalized local cohomology modules with respect to the class of modules in dimension less than a fixed integer","authors":"A. Vahidi, Saeid Morsali","doi":"10.36045/j.bbms.201230","DOIUrl":"https://doi.org/10.36045/j.bbms.201230","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"102 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79388267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A new approach to the three bisectors problem","authors":"D. Marinescu, Eugen Păltănea","doi":"10.36045/j.bbms.211105","DOIUrl":"https://doi.org/10.36045/j.bbms.211105","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"223 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80019727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mapping properties of the fractional integral operators on Herz-Hardy spaces with variable exponents","authors":"K. Ho","doi":"10.36045/j.bbms.211026","DOIUrl":"https://doi.org/10.36045/j.bbms.211026","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"2006 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88136233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Distinguished vector-valued continuous function spaces and injective tensor products","authors":"J. C. Ferrando, J. Ka̧kol","doi":"10.36045/j.bbms.210812","DOIUrl":"https://doi.org/10.36045/j.bbms.210812","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85592261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Jet schemes of quasi-homogeneous hypersurfaces and motivic monodromy conjecture for isolated quasi-homogeneous hypersurface singularities","authors":"J. Sebag","doi":"10.36045/j.bbms.210506","DOIUrl":"https://doi.org/10.36045/j.bbms.210506","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89241230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
of their size), the idea of shifting neighbourhoods of the group unit via the translation homeomorphisms do not work. Despite this difficulty, the paper proves that there are again induced uniformities (both of Tukey or Weil type) and the two are equivalent. The major tool in the paper is that of a special type of involutive quantale (called G -quantale ), which is borne out of the group structure of the locale. Such quantales are investigated and characterised paving the path to formulating proper-ties of the enriched quantales from which the group structure can be reconstructed. The localic group homomorphisms then turn out to be in one to one correspondence with quantale homomorphisms. Finally, the G -quantales are also ordered semigroups and hence can be used as a set of values for a generalised metric. The paper shows the natural group uniformities are metric uniformities of thus generalised metrics.
{"title":"Uniformities and a quantale structure on localic groups","authors":"J. Picado, A. Pultr","doi":"10.36045/j.bbms.210823","DOIUrl":"https://doi.org/10.36045/j.bbms.210823","url":null,"abstract":"of their size), the idea of shifting neighbourhoods of the group unit via the translation homeomorphisms do not work. Despite this difficulty, the paper proves that there are again induced uniformities (both of Tukey or Weil type) and the two are equivalent. The major tool in the paper is that of a special type of involutive quantale (called G -quantale ), which is borne out of the group structure of the locale. Such quantales are investigated and characterised paving the path to formulating proper-ties of the enriched quantales from which the group structure can be reconstructed. The localic group homomorphisms then turn out to be in one to one correspondence with quantale homomorphisms. Finally, the G -quantales are also ordered semigroups and hence can be used as a set of values for a generalised metric. The paper shows the natural group uniformities are metric uniformities of thus generalised metrics.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"91 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83347148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Completeness in topological vector spaces and filters on $mathbb N$","authors":"V. Kadets, D. Seliutin","doi":"10.36045/j.bbms.210512","DOIUrl":"https://doi.org/10.36045/j.bbms.210512","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"18 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84105744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Colorful Carathéodory, Helly and Sierksma numbers of convexity spaces","authors":"Krzysztof Kołodziejczyk","doi":"10.36045/j.bbms.210414","DOIUrl":"https://doi.org/10.36045/j.bbms.210414","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"127 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81818165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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