Obstruction Theory – a quick guide
In the first four parts we built a temporal world:
- Part 1 gave us a clean algebra of time – right‑open intervals, affine maps, and a global ordering.
- Part 2 showed how a discrete sampling function (S) is a perfect bijection with those intervals, so a finite list of samples contains exactly the same information as the continuous timeline (Theorem 2).
- Part 3 lifted the whole picture into the language of sheaves: to every open set (U) we attached the set of admissible “local sections’’ ( \mathcal F(U)) (the data we can actually observe on (U) ) together with restriction maps (\rho_{V!U}). The familiar gluing axiom said that if local sections agree on overlaps they can be glued into a global section.
All of this works as long as the local pieces really do glue. When they don’t, the failure is called an obstruction. In ordinary topology one meets the same idea when trying to extend a vector field, a line bundle, or a differential form: the obstruction lives in a cohomology group (most often (H^{1}) or (H^{2})).
Obstruction theory is precisely the study of when and how such gluing fails, and of the algebraic objects that measure the failure.
A time circuit (Section 6.1) is a network of temporal spaces linked by affine maps and sampled by apertures. Even if every single edge is perfectly invertible (no Type I–V obstruction), the composition of several edges can still destroy the ability to reconstruct the original signal. Two concrete mechanisms appear:
| Mechanism | What goes wrong | Obstruction type |
|---|---|---|
| Sampling mismatch – two successive samplers pick different phases (e.g. 30 fps → 24 fps → 30 fps) | The local pieces agree on the overlap but the phase of the underlying waveform is scrambled. | Type VI – aperture‑gluing smearing |
| Temporal collapse – a sensor integrates over a whole exposure (camera shutter) | All points of a time interval are sent to a single point; the temporal dimension is lost. | Type V – grade reduction |
These are global phenomena: they are invisible if you look at a single edge, but they appear as soon as you try to glue the edges together into a full circuit. Obstruction theory supplies the language to formalise exactly this “gluing‑fails’’ situation.
- Sheaf of temporal sections (\mathcal F) on a base space (\mathcal T) (the timeline).
- Local data: for each open interval (U\subset\mathcal T) we have a set (\mathcal F(U)) (samples, frames, pixel values …).
- Restriction maps (\rho_{VU}:\mathcal F(U)\to\mathcal F(V)) for (V\subseteq U).
Given a cover ({U_i}) of (\mathcal T) we obtain a Čech cochain complex
[ 0;\longrightarrow;\prod_i!\mathcal F(U_i) \xrightarrow{;\delta^0;}! \prod_{i<j}!\mathcal F(U_i\cap U_j) \xrightarrow{;\delta^1;}! \prod_{i<j<k}!\mathcal F(U_i\cap U_j\cap U_k) ;\longrightarrow;\cdots ]
The first cohomology group
[ H^{1}(\mathcal T,\mathcal F)=\frac{\ker\delta^{1}}{\operatorname{im}\delta^{0}} ]
contains exactly those compatibility classes of local sections that cannot be glued to a global one. A non‑zero class is an obstruction.
- (H^{0}) non‑trivial – there is more than one global section (multiple possible “worlds’’).
- (H^{1}\neq0) – the local data are mutually compatible on pairwise overlaps but no global section exists; the failure is measured by a cohomology class.
In the temporal setting the type of the class is determined by the physical cause:
| Type | Origin | Cohomology description |
|---|---|---|
| I | Insufficient sampling rate | (\mathcal O_{I}=0) iff Nyquist satisfied; otherwise the kernel of the sampling map is non‑trivial (loss of high‑frequency modes). |
| II | Scale/phase mismatch | A non‑trivial 0‑cocycle (a global scaling factor (S\neq1)) that cannot be undone locally. |
| III | Bounded aperture (temporal window) | (\mathcal O_{III}\in H^{1}) – the “hole’’ created by the missing part of the timeline. |
| IV | Orientation reversal (sign‑flip) | A (\mathbb Z_{2})‑valued 1‑cocycle (the orientation bundle). |
| V | Grade reduction ((S=0)) | Collapse of a dimension; the sheaf lives on a lower‑dimensional base. |
| VI | Aperture‑gluing smearing (phase‑mixing) | A higher obstruction that appears only after composing several maps; it lives in (H^{1}) but is proportional to the phase‑entropy produced by the circuit (Theorem 6.3). |
Thus each physical limitation appears as a precise algebraic class.
| Part | Main tool | How obstruction theory uses it |
|---|---|---|
| 1 | Temporal algebra (right‑open intervals, affine maps) | Supplies the base space (\mathcal T) and the transition maps (\pi) that generate the sheaf. |
| 2 | Sampling theory (bijection between discrete indices and intervals) | Gives the local sections (\mathcal F(U)) (samples) and tells us when the sampling map is an isomorphism (no Type I). |
| 3 | Sheaf theory (restriction, gluing) | Provides the cochain complex whose cohomology detects failures of gluing – the core of obstruction theory. |
| 4 | Sheaf‑pebbling synthesis (five obstruction types) | Identifies the concrete physical meanings of the cohomology classes (Types I–V). |
| 5 | Oriented charts (allowing (S<0)) | Extends the sheaf to a bundle with orientation; the non‑trivial orientation cocycle is the Type IV obstruction. |
| 6 | Time circuits (composition of edges) | Shows that new obstructions (Type VI) arise only when multiple edges are composed – a higher‑order cohomological effect. |
Obstruction theory is therefore the unifying language that ties together the algebraic, sampling, sheaf‑theoretic, and circuit‑level structures introduced earlier. It tells us exactly what is lost when we pass from a perfectly coherent global timeline to the fragmented, sampled reality we can actually observe.
Take a pure sinusoid (s(t)=\cos(\omega t)) on the interval ([0,1]).
Edge E₁ samples it at 30 fps (no obstruction).
Edge E₂ resamples at 24 fps.
Individually each edge has ( \mathcal O_{I}= \mathcal O_{II}= \mathcal O_{III}=0). Yet the composite map (\pi_{2}!\circ!\pi_{1}) sends the 30‑fps samples to a set of points that are out of phase with the 24‑fps grid. The Čech 1‑cocycle measuring the mismatch on overlaps is non‑zero; that is precisely Type VI. The resulting reconstructed signal is blurred, and the obstruction class tells us how much phase entropy has been injected (Theorem 6.3).
Obstruction theory answers the central question of the book:
Given a locally coherent temporal world, why does a bounded observer ever see a “flow’’ of time, a collapse of a wavefunction, or an irreversible loss of information?
The answer is that the observer’s apertures, sampling rates, orientations, and any grade reductions generate cohomology classes. When those classes are non‑trivial, global coherence cannot be reconstructed; the resulting irreversibility is the arrow of time we experience. All the machinery built in Parts 1‑5 is precisely what we need to write down, compute, and ultimately test those cohomology classes.