Homotopy in enriched functor categories: Frobenius exact structures and weak Serre functors
Abstract
Let (V, ⊗, I,[−,−]) be a closed symmetric monoidal category with finite (co)limits. Assume that V is weakly idempotent complete and additive. For a given small V-category A, we investigate sufficient conditions on A under which the ordinary category [A, V]0 of V-functors from A to V, equipped with the exact structure ℰlowt, is a Frobenius exact category,and hence, induces a triangulated stable category. The exact structure ℰlowt encompasses the degre-wise trivial exact structure on categories of N-chain complexes and m-periodic chain complexes, with terms in a variety of settings, including left R-modules, quasi-coherent sheaves on a scheme, and Banach spaces. We subsequently address the question of describing abstract left and right homotopy relations in ([A, V]0;ℰlowt) in terms of families of morphisms defined objectwise, as in the cases of chain-homotopy and relative homotopy relations. To this end, we introduce the notion of left/right-weak Serre V-functors on A, and investigate their relationships with the classical Serre functor relative to a commutative ring k.
Related
- Frobenius functors and one-sided Hopf algebras in braided monoidal categories
- Transfer of abelian model structures to equivariant categories and homotopy squares
- Compact generation and perfect objects in homotopy categories of exact model structures
- Frobenius functors and n-torsionfree objects
- Functors from Lagrangian correspondences for wrapped sheaf categories
BibTeX
@misc{odabasi2026homotopy,
title = {{Homotopy in enriched functor categories: Frobenius exact structures and weak Serre functors}},
author = {Sinem Odabasi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40746},
eprint = {2610.40746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40746},
}
RIS
TY - UNPB
TI - Homotopy in enriched functor categories: Frobenius exact structures and weak Serre functors
AU - Sinem Odabasi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40746
UR - https://arxiv.org/abs/2610.40746
AB - Let (V, ⊗, I,[−,−]) be a closed symmetric monoidal category with finite (co)limits. Assume that V is weakly idempotent complete and additive. For a given small V-category A, we investigate sufficient conditions on A under which the ordinary category [A, V]_0 of V-functors from A to V, equipped with the exact structure ℰ_lowt, is a Frobenius exact category,and hence, induces a triangulated stable category. The exact structure ℰ_lowt encompasses the degre-wise trivial exact structure on categories of N-chain complexes and m-periodic chain complexes, with terms in a variety of settings, including left R-modules, quasi-coherent sheaves on a scheme, and Banach spaces. We subsequently address the question of describing abstract left and right homotopy relations in ([A, V]_0;ℰ_lowt) in terms of families of morphisms defined objectwise, as in the cases of chain-homotopy and relative homotopy relations. To this end, we introduce the notion of left/right-weak Serre V-functors on A, and investigate their relationships with the classical Serre functor relative to a commutative ring k.
ER -