Geodesic
Homotopy equivalence of normalized and unnormalized complexes, revisited
Algebra and discrete mathematics, Tome 32 (2021) no. 2, pp. 253-266

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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, Dold–Kan correspondence.
Mots-clés : simplicial object; homotopy in chain complexes 
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     author = {V. Lyubashenko and A. Matsui},
     title = {Homotopy equivalence of normalized and unnormalized complexes, revisited},
     journal = {Algebra and discrete mathematics},
     pages = {253--266},
     publisher = {mathdoc},
     volume = {32},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a7/}
}
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V. Lyubashenko; A. Matsui. Homotopy equivalence of normalized and unnormalized complexes, revisited. Algebra and discrete mathematics, Tome 32 (2021) no. 2, pp. 253-266. http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a7/