Chromatic Obstruction Sheaf II - The Fundamental Obstruction Sheaf

Abstract
We construct the fundamental obstruction object of Chromatic Obstruction Sheaf Theory
within the relational descent environment developed in COST I. Starting from a small
finitely complete relational site (C, J) and its hypercomplete ∞-topos X = Shvhyp
S (C, J),
we define an intrinsic finite-stage obstruction functor by measuring pointed relative de-
fects associated to hypercover realizations. These defects are represented by finite relative
cofiber objects, and their colimit produces a canonical pointed object
Ofund ∈ X∗,
the fundamental unstable obstruction carrier. We prove that Ofund represents the rela-
tional obstruction functor and is therefore determined by the descent structure up to
contractible equivalence.
We then pass to the stable spectral image
Ffund := Σ∞Ofund ∈ Sp(X ),
without identifying this stable image with the original unstable carrier. For connective
Ffund, the Postnikov tower yields stable coefficient sheaves Afund,n := πn(Ffund), primary
stable obstruction classes
ost
fund,n(s) ∈ Hn+1(U ; Afund,n),
and associated obstruction sheaves obtained by sheafifying the corresponding local coho-
mology presheaves. These classes satisfy the usual vanishing and torsor criteria for lifting
through successive Postnikov stages.
Finally, we organize the stable obstruction package by chromatic localization, height
filtration, fracture reconstruction, base change, and admissible Morita invariance. Chro-
matic localization is applied at the stable level, producing height-filtered coefficient and
obstruction sheaves, while fracture data give Mayer–Vietoris compatibility conditions
for reconstructing obstruction classes from height pieces. Under explicit compatibility
hypotheses, the fundamental obstruction carrier, its stable image, coefficient sheaves, ob-
struction classes, chromatic localizations, and fracture data are invariant under admissible
changes of relational presentation. Thus COST II supplies the canonical obstruction-
theoretic core of the theory: an unstable universal carrier together with its stable chro-
matic obstruction package, forming the foundation for subsequent directive, moduli, and
cotangent constructions.