1. Scope and contribution
An observation usually leaves a family of compatible hidden states. In a higher categorical model this family also contains paths, automorphisms, and higher coherence. We call the study of this residual structure fiber observability. The term specifies a framework in this paper; no claim is made that it denotes a previously standardized theory.
Classical nonlinear observability distinguishes states through input-output behavior [1]. Coalgebra organizes behavioral equivalence and observable quotients [2]. Higher topos theory and homotopy type theory provide fibers, truncation, and descent [3,4]. Statistical experiments distinguish states through output laws and decision risks [5-7]. We combine these ingredients into one typed account, with explicit proofs and finite examples.
The contribution is a synthesis: a relative definition of coherent observability; a probe-resolution obstruction; tests using generators and descent; finite-horizon reconstruction; and statistical impossibility and confidence bounds. Several ingredients are standard facts or elementary consequences. Their roles and assumptions are kept explicit; the framework does not supply a general algorithm for arbitrary higher objects.
2. Observations and their coherent fibers
Fix an infinity-topos $\mathcal E$ and work in a universe permitting the small limits used below. A known context is an object $B$. A hidden-state object $p:X\to B$ and a record object $q:Y\to B$ belong to the slice $\mathcal E_{/B}$. An observer is a morphism $o:X\to Y$ in that slice. Products and equalities below are homotopy-coherent unless we explicitly work with sets.
Definition 2.1 (observation fiber). For a generalized record $y:T\to Y$, define
$$ F_y=X\times_Y T\quad\text{in }\mathcal E_{/T}. $$
Internally, over a record $y$, the fiber has the type
$$ F_y\simeq\sum_{x:X}(o(x)=_Y y). $$
Thus a realization includes a state and a witness that its observation matches the record. An inhabited fiber expresses realizability; a subterminal fiber expresses at most one coherent realization; a contractible fiber expresses exactly one. A subterminal object has its map to the terminal object a monomorphism. It need not be globally inhabited.
Definition 2.2 (full fiber observability). The observer is fully fiber-observable if $o$ is a monomorphism, equivalently a $(-1)$-truncated map. Its image factorization is
$$ X\xrightarrow{e}I\xrightarrow{j}Y, $$
where $e$ is an effective epimorphism and $j$ is a monomorphism. The object $I$ records supported observations. An effective epimorphism is a map whose augmented Cech nerve has its codomain as colimit.
Proposition 2.3 (unique reconstruction on the image). The following are equivalent: (i) $o$ is fully fiber-observable; (ii) its diagonal $X\to X\times_YX$ is an equivalence; (iii) every generalized fiber is subterminal; (iv) $e:X\to I$ is an equivalence.
*Proof.* The diagonal criterion is the definition of a monomorphism in an infinity-category with pullbacks. Base change gives the equivalence with subterminal fibers, including the test $T=Y$. If $o=je$ is mono, then $e$ is mono; an effective epimorphism that is mono is an equivalence. Conversely an equivalence followed by $j$ is mono. $\square$
Consequently reconstruction from $I$ is unique up to a contractible space of choices. Records outside the image need not have realizations. For sets this is ordinary injectivity. For spaces it is stronger than injectivity on connected components: compatibility witnesses are part of the chosen semantics.
Example 2.4 (hidden symmetry). Let $G$ be a nontrivial discrete group and $BG$ its one-object classifying groupoid. The map $BG\to *$ has one connected component in its fiber, but its fiber has loop group $G$ and is not subterminal. Conversely $*\to BG$ has one underlying state, yet its homotopy fiber at the basepoint is $G$: there are multiple compatibility witnesses. These examples prevent interpreting a homotopy fiber as only a set of state labels.
3. Probe refinement and a truncation barrier
An additional probe $a:X\to Z$ over $B$ gives the joint observer
$$ Q=(o,a):X\longrightarrow Y\times_B Z. $$
Theorem 3.1 (residual fiber under a new probe). For compatible records $y:T\to Y$ and $z:T\to Z$, let $Z_T=Z\times_BT$ and let $a_y:F_y\to Z_T$ be induced by $a$. Then
$$ \operatorname{fib}_{(y,z)}(Q)\simeq F_y\times_{Z_T}T_z. $$
Here $T_z\to Z_T$ is the section given by $z$. Joint observation is fully fiber-observable exactly when these residual fibers are subterminal for all generalized records.
*Proof.* Both sides are limits of the diagram imposing $o(x)=y$ and $a(x)=z$ over their common context. Associativity of pullbacks identifies the limits, with all compatibility data retained. The final assertion is Proposition 2.3. $\square$
The theorem describes which ambiguity a probe removes. Adding a probe can resolve states, preserve a residual symmetry, or reveal that a proposed joint record has no realization.
We next bound probe resolution. For $n\ge0$, an object is $n$-truncated when every mapping space into it has no homotopy groups above degree $n$. A morphism is $n$-truncated when it is an $n$-truncated object in its codomain slice. Relative truncation $\tau_{\le n}^{Y}$ is the left adjoint to inclusion of these objects in $\mathcal E_{/Y}$ [3, Proposition 5.5.6.18].
Theorem 3.2 (probe-resolution barrier). Regard $o:X\to Y$ as an object in $\mathcal E_{/Y}$. Let $r_i:X\to Z_i$ be a small family of probes over $Y$, where every $Z_i\to Y$ is $n$-truncated. The joint probe factors, uniquely up to contractible choice, as
$$ X\longrightarrow\tau_{\le n}^{Y}X \longrightarrow\prod_{i,Y}Z_i. $$
If the joint probe is a monomorphism, then $X\to Y$ is $n$-truncated. Conversely, if $X\to Y$ is $n$-truncated, the unrestricted class of such probes contains a fully observing probe.
*Proof.* The product is $n$-truncated, since truncated objects are closed under limits. The truncation adjunction gives the factorization and its uniqueness. If $r:X\to\prod_{i,Y}Z_i$ is mono, its composite with the product's structure map is $n$-truncated: composition of a $(-1)$-truncated map and an $n$-truncated map is $n$-truncated. This composite is $o$. For the converse choose $Z=X$ and $r=\mathrm{id}_X$. $\square$
The converse is an expressiveness statement, not an implementable sensor construction. In spaces, $\tau_{\le0}BG\simeq *$, so every set-valued probe on a fixed $BG$ is constant on its internal symmetry. More generally $K(A,n+1)$, for a nonzero abelian group $A$, is invisible beyond its terminal $n$-truncation to such probes. A scalar invariant defined on a different moduli space may distinguish components representing different groups; that does not contradict the theorem about a specified domain.
4. Adequate tests, locality, and descent
Global states alone need not detect observability in a topos. Call a family $\mathcal G$ of test objects conservative when the functors $\operatorname{Map}(G,-)$, $G\in\mathcal G$, jointly detect equivalences.
Theorem 4.1 (test adequacy). For a conservative family, $o:X\to Y$ is mono if and only if
$$ \operatorname{Map}(G,X)\longrightarrow\operatorname{Map}(G,Y) $$
is a monomorphism of spaces for every $G\in\mathcal G$.
*Proof.* Mapping out of $G$ preserves limits, so a mono gives a mono. Conversely these conditions make the images of the diagonal $X\to X\times_YX$ equivalences. Conservativity makes the diagonal an equivalence. $\square$
The test uses whole mapping spaces. Checking injectivity of their $\pi_0$ alone fails even in spaces, as $BG\to *$ shows.
Theorem 4.2 (locality). If $u:U\to Y$ is an effective epimorphism, then $o:X\to Y$ is mono if and only if $o_U:X\times_YU\to U$ is mono.
*Proof.* One direction is stability under pullback. For the other, the pullback of the diagonal of $o$ is the diagonal of $o_U$, hence an equivalence. Pullback along an effective epimorphism is conservative: equivalences descend along its Cech nerve [3, Lemma 6.2.3.16]. Therefore the original diagonal is an equivalence. $\square$
Locality of this property must be distinguished from the existence of chosen global states. In the infinity-topos of sheaves of spaces on $S^1$, let $P$ be the sheaf of local sections of the double cover $z\mapsto z^2$. The map $P\to1$ is locally a two-element set over $1$, hence effective epi and not mono. It has no global section: such a section $s:S^1\to S^1$ would satisfy $2\deg(s)=1$. Nevertheless the global-sections map $\varnothing\to\{*\}$ is injective. Global sections therefore miss the failure of observability and local choices need not glue.
5. Dynamic observation and finite horizons
Let $X$ be a nonempty finite set of size $N$, $A$ a finite input alphabet, $\delta_a:X\to X$ transitions, and $h:X\to O$ an output function. For a word $w=a_1\cdots a_k$, put $\delta_w=\delta_{a_k}\cdots\delta_{a_1}$, with $\delta_{\varnothing}=\mathrm{id}$. Define
$$ Q_L(x)=\bigl(h(\delta_wx)\bigr)_{|w|\le L}, \qquad R_L=\ker(Q_L). $$
The kernel here is an ordinary equivalence relation. The protocol assumes access to the indicated responses from the same initial state, for example by resetting the system. It is not automatically a single unresettable experiment.
Theorem 5.1 (finite stabilization). The relations satisfy
$$ R_{L+1}=R_0\cap\bigcap_{a\in A} (\delta_a\times\delta_a)^{-1}(R_L). $$
They stabilize by $L=N-1$. The limit $R_\infty$ is the largest transition congruence contained in $\ker(h)$. All states are distinguishable by finite words exactly when $Q_{N-1}$ is injective.
*Proof.* Every nonempty word of length at most $L+1$ is a first input followed by a word of length at most $L$, giving the recurrence. The sequence refines partitions. Each strict refinement increases the number of blocks, which is at most $N$. If one step is equal, the recurrence makes all later steps equal; thus there cannot be $N$ strict steps. At stability the relation lies in $\ker(h)$ and is preserved by every transition. Conversely any such congruence identifies states with equal output after every word, by induction, so is contained in every $R_L$. The last assertion follows. $\square$
This is the finite deterministic specialization of behavioral observability [2]. For $X=\{0,1\}^2$, $h(a,b)=a$, and swap transition $\delta(a,b)=(b,a)$, $Q_0$ has two states per fiber while $Q_1$ distinguishes all four. With the identity transition, the second bit remains hidden forever.
Finite-state stabilization has no unrestricted infinite-state analogue. On $X=\{0,1\}^{\mathbb N}$, all coordinate probes jointly determine a sequence, while any finite subfamily leaves infinitely many coordinates free. These probes are continuous for the compact Cantor topology; compactness of the state space does not force a finite separating subfamily.
6. Noisy probes and identification from laws
Let $X,Y$ be finite sets. A stochastic observer is a channel $K:X\rightsquigarrow Y$, assigning a probability vector $K_x$ to each state. Its law observer is $\kappa:X\to\Delta(Y)$, $\kappa(x)=K_x$, where $\Delta(Y)$ is regarded as a set of laws, not a topological simplex. Equal-law fibers differ from fibers of a realized sample. Markov categories describe channel composition [6,7]; the quantitative assumptions below are additional.
Write $\mathrm{TV}(P,Q)=\tfrac12\sum_y|P(y)-Q(y)|$.
Theorem 6.1 (testing obstruction and garbling). For two states with laws $P,Q$, any randomized test based on $m$ conditionally independent samples has equal-prior error at least
$$ \frac{1-\mathrm{TV}(P^{\otimes m},Q^{\otimes m})}{2}. $$
Equality is achieved by a likelihood-ratio test. Consequently equal-law states cannot be uniformly identified with error below $1/2$ by any finite number of these samples. Postprocessing by another channel cannot remove this obstruction.
*Proof.* At a sample string $s$, deciding for either hypothesis contributes at least $\tfrac12\min(P^{\otimes m}(s),Q^{\otimes m}(s))$ to the error; randomized decisions cannot improve a pointwise minimum. Summing uses $\sum_s\min(P(s),Q(s))=1-\mathrm{TV}(P,Q)$. Choose the more likely hypothesis at each string to attain equality. If $P=Q$, their powers coincide. For a postprocessing channel $M$, the triangle inequality gives
$$ \tfrac12\sum_z\left|\sum_y(P(y)-Q(y))M(z\mid y)\right| \le\tfrac12\sum_y|P(y)-Q(y)|. $$
Thus total variation cannot increase, and equal laws remain equal. $\square$
This elementary obstruction is consistent with the comparison-of-experiments viewpoint [5,7]. It concerns the declared experiment; a new intervention or a new channel may distinguish states that this channel does not.
Theorem 6.2 (finite confidence bound). Suppose $|X|\ge2$, $M=|Y|<\infty$, the channel is known, and
$$ \Delta=\min_{x\ne x'}\|K_x-K_{x'}\|_\infty>0. $$
From $m$ independent samples, let $\widehat p$ be the empirical distribution and choose a state minimizing $\|\widehat p-K_x\|_\infty$, with a fixed rule for ties. Uniformly over the true state,
$$ \Pr(\widehat x\ne x)\le2M\exp(-m\Delta^2/2). $$
Hence $m\ge(2/\Delta^2)\log(2M/\delta)$ suffices for confidence $1-\delta$.
*Proof.* If $\|\widehat p-K_x\|_\infty<\Delta/2$, the triangle inequality makes every other law farther from $\widehat p$. A union bound over the $M$ coordinates, using Hoeffding's inequality [8] for their indicator variables, bounds the complement by $2M e^{-2m(\Delta/2)^2}$. $\square$
This is a finite identification guarantee for known, separated laws. For Bernoulli parameters $1/4$ and $3/4$, $M=2$ and $\Delta=1/2$; $m=36$ suffices for $\delta=0.05$ by this bound. Homotopical observability alone does not give a separation margin, independent samples, or a statistical rate.
7. Task recovery and coherent descent
Reconstructing a whole state may exceed what a task requires. For sets and $I=\operatorname{im}(o)$, a task $g:X\to Z$ is recoverable from records if $g=\bar g e$ for some $\bar g:I\to Z$.
Proposition 7.1 (set-valued task criterion). Such a $\bar g$ exists exactly when $g$ is constant on every fiber of $e$; if it exists it is unique.
*Proof.* A factorization is constant on fibers. Conversely define $\bar g(y)$ as the common value of $g$ on the nonempty fiber $e^{-1}(y)$. This value is uniquely determined, so no arbitrary representative choice enters the definition. Surjectivity gives uniqueness. $\square$
In an infinity-topos, let $X_\bullet$ be the Cech nerve of the effective epi $e:X\to I$, so $X_k=X\times_I\cdots\times_IX$ with $k+1$ factors.
Theorem 7.2 (coherent task recovery). For every $Z$,
$$ \operatorname{Map}(I,Z)\simeq \lim_{[k]\in\Delta}\operatorname{Map}(X_k,Z). $$
Here $\Delta$ is the simplex category. A task $g:X\to Z$ extends precisely when it admits the descent data on the right with degree-zero component $g$. Its reconstruction space is
$$ \operatorname{Rec}_e(g)= \operatorname{fib}_g\bigl(\operatorname{Map}(I,Z)\to\operatorname{Map}(X,Z)\bigr). $$
*Proof.* Effective epimorphy identifies $I$ with the colimit of $X_\bullet$. Mapping this colimit into $Z$ gives the stated limit. Taking the fiber over the prescribed degree-zero component gives exactly the extensions with their compatibility witness. $\square$
The descent data include a homotopy between the two pullbacks of $g$ to $X_1$ and compatible higher homotopies. Once complete descent data are specified, the corresponding realization is unique up to contractible choice. Specifying $g$ alone need not specify those data.
For example, $e:*\to BG$ is an effective epi when $G$ is a discrete group. The basepoint task $g:*\to BG$ admits both the identity extension $BG\to BG$ and the constant extension. For nontrivial $G$ they differ on $\pi_1$. Thus effective epimorphy alone does not make a higher-valued extension unique. Conversely the task $\mathrm{id}_{BG}$ does not factor through $BG\to *$: such a factorization would kill its nontrivial fundamental group.
For a binary task on finite stochastic states, if an equal-law fiber contains different task labels, Theorem 6.1 gives the same $1/2$ worst-case error obstruction for that task. Exact state reconstruction can therefore fail while a task constant on the law fibers remains identifiable.
8. Interpretation and limits
The framework specifies four obligations: a context and observation type; residual coherent fibers; an admissible probe or intervention family; and, for statistical claims, an experiment with quantitative assumptions. Full fiber observability is relative to those choices. Truncating the chosen equality can legitimately weaken the reconstruction demand, but the discarded symmetries then lie outside that demand.
For neural representations or distributed agents, an observation might record predictions, traces, or local compatibility tests. The results apply after these are represented by the stated maps or finite channels. They do not imply that architecture, state dimension, or a list of scalar scores determines an observation fiber without constructing the observation map. Finite simulations illustrate the discrete statements; they do not verify the general infinity-topos results.
Further work can add costs, adaptive interventions, effective presentations, or quantitative coherence bounds to this specification.
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*Authorship and evidence.* Anonymous author. Prepared with AI assistance. Proofs are mathematical arguments from the stated assumptions and cited standard foundations; they are not proof-assistant certified. The accompanying code checks finite examples with exact arithmetic. No empirical validation or priority claim for the standard constituent theorems is asserted.