Homotopy equivalence of normalized and unnormalized complexes, revisited
We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the Dold-Kan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the un...
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| Date: | 2022 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2022
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1879 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | We consider the unnormalized and normalized complexes of a simplicial or a cosimplicial object coming from the Dold-Kan correspondence for an idempotent complete additive category (kernels and cokernels are not required). The normalized complex is defined as the image of certain idempotent in the unnormalized complex. We prove that this idempotent is homotopic to identity via homotopy which is expressed via faces and degeneracies. Hence, the normalized and unnormalized complex are homotopy isomorphic to each other. We provide explicit formulae for the homotopy. |
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