homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Integral homotopy theory is the refinement of rational homotopy theory to integer coefficients.
Just as there are two established approaches in rational homotopy theory for encoding rational homotopy types, those of Quillen and of Sullivan, so there are analog approaches for integral homotopy types. In Blomquist & Harper 16, chains in ordinary cohomology with rational number coefficients are lifted to chains with integer coefficients. While Horel 2022) and (Yuan 23 both employ cochains.
Integral analogs of dg-algebraic rational homotopy theory equivalence:
Geoffroy Horel: Binomial rings and homotopy theory, Journal für die reine und angewandte Mathematik 2024 813 (2024) [arXiv:2211.02349, doi:10.1515/crelle-2024-0039]
Allen Yuan: Integral models for spaces via the higher Frobenius, Journal of the American Mathematical Society 36 1 (2023) 107–175 [arXiv:1910.00999, doi:10.1090/jams/998]
Last revised on February 23, 2026 at 07:03:20. See the history of this page for a list of all contributions to it.