Pub Date : 2020-04-13DOI: 10.52547/cgasa.2023.103289
P. P. Ghosh
Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in my earlier paper. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of continuity of morphisms with respect to induced closure operators.
{"title":"Internal Neighbourhood Structures II: Closure and closed morphisms","authors":"P. P. Ghosh","doi":"10.52547/cgasa.2023.103289","DOIUrl":"https://doi.org/10.52547/cgasa.2023.103289","url":null,"abstract":"Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in my earlier paper. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of continuity of morphisms with respect to induced closure operators.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41312519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.
{"title":"Six model categories for directed homotopy","authors":"P. Gaucher","doi":"10.52547/cgasa.15.1.145","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.145","url":null,"abstract":"We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43689662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
These notes give an elementary and formal 2-categorical construction of the bicategory of anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful.
{"title":"The elementary construction of formal anafunctors","authors":"David Michael Roberts","doi":"10.52547/cgasa.15.1.183","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.183","url":null,"abstract":"These notes give an elementary and formal 2-categorical construction of the bicategory of anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48622403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,varepsilon,theta)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.
{"title":"Simplicial structures over the 3-sphere and generalized higher order Hochschild homology","authors":"Samuel Carolus, J. Laubacher","doi":"10.52547/cgasa.15.1.93","DOIUrl":"https://doi.org/10.52547/cgasa.15.1.93","url":null,"abstract":"In this paper we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,varepsilon,theta)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.","PeriodicalId":41919,"journal":{"name":"Categories and General Algebraic Structures with Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44479695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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