Cohomology of Presheaves of Monoids
The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids w...
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| Format: | Article |
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MDPI AG
2020-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/1/116 |
| _version_ | 1811193022124130304 |
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| author | Pilar Carrasco Antonio M. Cegarra |
| author_facet | Pilar Carrasco Antonio M. Cegarra |
| author_sort | Pilar Carrasco |
| collection | DOAJ |
| description | The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth. |
| first_indexed | 2024-04-12T00:00:12Z |
| format | Article |
| id | doaj.art-1fa4aa988d2a416182a5b891e124c3ff |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-12T00:00:12Z |
| publishDate | 2020-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-1fa4aa988d2a416182a5b891e124c3ff2022-12-22T03:56:14ZengMDPI AGMathematics2227-73902020-01-018111610.3390/math8010116math8010116Cohomology of Presheaves of MonoidsPilar Carrasco0Antonio M. Cegarra1Department Algebra, University of Granada, 18071 Granada, SpainDepartment Algebra, University of Granada, 18071 Granada, SpainThe purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi> </semantics> </math> </inline-formula>-extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth.https://www.mdpi.com/2227-7390/8/1/116cohomologypresheaf of monoidsmonoidal prestacksimplicial sethomotopy colimit |
| spellingShingle | Pilar Carrasco Antonio M. Cegarra Cohomology of Presheaves of Monoids Mathematics cohomology presheaf of monoids monoidal prestack simplicial set homotopy colimit |
| title | Cohomology of Presheaves of Monoids |
| title_full | Cohomology of Presheaves of Monoids |
| title_fullStr | Cohomology of Presheaves of Monoids |
| title_full_unstemmed | Cohomology of Presheaves of Monoids |
| title_short | Cohomology of Presheaves of Monoids |
| title_sort | cohomology of presheaves of monoids |
| topic | cohomology presheaf of monoids monoidal prestack simplicial set homotopy colimit |
| url | https://www.mdpi.com/2227-7390/8/1/116 |
| work_keys_str_mv | AT pilarcarrasco cohomologyofpresheavesofmonoids AT antoniomcegarra cohomologyofpresheavesofmonoids |