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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100601/type/journal_article |
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| Summary: | 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. |
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| ISSN: | 2050-5094 |