Posets arising from decompositions of objects in a monoidal category
Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $ , where the neutral element for the product is an initial object, we consider the poset of $\sqcup $ -complemented subobjects of a given object X. When this poset has finite height, we define decompositions...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100601/type/journal_article |
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| author | Kevin Ivan Piterman Volkmar Welker |
| author_facet | Kevin Ivan Piterman Volkmar Welker |
| author_sort | Kevin Ivan Piterman |
| collection | DOAJ |
| description | Given a symmetric monoidal category
${\mathcal C}$
with product
$\sqcup $
, where the neutral element for the product is an initial object, we consider the poset of
$\sqcup $
-complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with
$\sqcup $
, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness. |
| format | Article |
| id | doaj-art-9e28cca82bf448c494a85cea67a1dd2c |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-9e28cca82bf448c494a85cea67a1dd2c2025-08-20T03:56:00ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10060Posets arising from decompositions of objects in a monoidal categoryKevin Ivan Piterman0https://orcid.org/0000-0002-9024-5698Volkmar Welker1https://ror.org/01rdrb571 Philipps-Universität Marburg , Fachbereich Mathematik und Informatik, 35032 Marburg, Germany; E-mail: https://ror.org/006e5kg04 Vrije Universiteit Brussel, Department of Mathematics and Data Science, 1050 Brussels, Belgium;https://ror.org/01rdrb571 Philipps-Universität Marburg , Fachbereich Mathematik und Informatik, 35032 Marburg, Germany; E-mail:Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $ , where the neutral element for the product is an initial object, we consider the poset of $\sqcup $ -complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with $\sqcup $ , and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness.https://www.cambridge.org/core/product/identifier/S2050509425100601/type/journal_article05E4520F6557M0706A07 |
| spellingShingle | Kevin Ivan Piterman Volkmar Welker Posets arising from decompositions of objects in a monoidal category Forum of Mathematics, Sigma 05E45 20F65 57M07 06A07 |
| title | Posets arising from decompositions of objects in a monoidal category |
| title_full | Posets arising from decompositions of objects in a monoidal category |
| title_fullStr | Posets arising from decompositions of objects in a monoidal category |
| title_full_unstemmed | Posets arising from decompositions of objects in a monoidal category |
| title_short | Posets arising from decompositions of objects in a monoidal category |
| title_sort | posets arising from decompositions of objects in a monoidal category |
| topic | 05E45 20F65 57M07 06A07 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100601/type/journal_article |
| work_keys_str_mv | AT kevinivanpiterman posetsarisingfromdecompositionsofobjectsinamonoidalcategory AT volkmarwelker posetsarisingfromdecompositionsofobjectsinamonoidalcategory |