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RESEARCH ARTICLE
Homotopy equivalence of normalized and unnormalized complexes, revisited
V. Lyubashenkoa, A. Matsuib a Institute of Mathematics NASU, 3 Tereshchenkivska st., Kyiv, 01024, Ukraine
b Kyiv National Taras Shevchenko University, Faculty of Mechanics and Mathematics, 4-e Akademika Hlushkova Ave, Kyiv, 03127, Ukraine
Abstract:
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.
Keywords:
idempotent, simplicial object; homotopy in chain complexes, Dold–Kan correspondence.
Received: 15.08.2021
Citation:
V. Lyubashenko, A. Matsui, “Homotopy equivalence of normalized and unnormalized complexes, revisited”, Algebra Discrete Math., 32:2 (2021), 253–266
Linking options:
https://www.mathnet.ru/eng/adm820 https://www.mathnet.ru/eng/adm/v32/i2/p253
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Abstract page: | 71 | Full-text PDF : | 46 | References: | 11 |
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