The first Cech derived functor — Draconitic Research

Abstract

Let 𝒞 be a semi-abelian category, 𝒟 a Birkhoff subcategory of 𝒞, and I the associated reflector. For C∈ 𝒞, we establish an isomorphism between H₁(N(I(checkC(f,C)))) (where N is the Moore normalization functor and checkC(f,C) is a Cech resolution associated to C) and H₁(N(I(𝔾(C)))) (where 𝔾(C) is the canonical 𝔾-simplicial resolution of C for 𝔾 the comonad derived from an adjunction F: 𝒳→ 𝒞dashv U: 𝒞→ 𝒳 satisfying classical conditions introduced by Quillen). The former corresponds to the Hopf formula for the second homology object associated to I, while the latter represents the first \v Cech derived functor of I. Specifically, we show that both can be expressed by a single unified formula. Furthermore, we can also prove that, under certain assumptions, the first \v Cech derived functor of I is naturally isomorphic to the first non-additive derived functor of I (in the sense of Culot--Renaud--Van~der~Linden).

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BibTeX RIS

BibTeX

@misc{culot2026first,
  title = {{The first Cech derived functor}},
  author = {Maxime Culot and Tomas Everaert},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24720},
  eprint = {2610.24720},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24720},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The first Cech derived functor
AU  - Maxime Culot
AU  - Tomas Everaert
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24720
UR  - https://arxiv.org/abs/2610.24720
AB  - Let 𝒞 be a semi-abelian category, 𝒟 a Birkhoff subcategory of 𝒞, and I the associated reflector. For C∈ 𝒞, we establish an isomorphism between H_1(N(I(checkC(f,C)))) (where N is the Moore normalization functor and checkC(f,C) is a Cech resolution associated to C) and H_1(N(I(𝔾(C)))) (where 𝔾(C) is the canonical 𝔾-simplicial resolution of C for 𝔾 the comonad derived from an adjunction F: 𝒳→ 𝒞dashv U: 𝒞→ 𝒳 satisfying classical conditions introduced by Quillen). The former corresponds to the Hopf formula for the second homology object associated to I, while the latter represents the first \v Cech derived functor of I. Specifically, we show that both can be expressed by a single unified formula. Furthermore, we can also prove that, under certain assumptions, the first \v Cech derived functor of I is naturally isomorphic to the first non-additive derived functor of I (in the sense of Culot--Renaud--Van~der~Linden).
ER  -

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