lin-0218

18.1 Compatibility and effectivity

Let \(\{ U_i\to U\} \) be a finite cover and let a presheaf \(X\) assign a predictive-state object \(X(U_i)\) to each context. Restrictions expose shared questions:

\[ \rho ^i_{ij}:X(U_i)\longrightarrow X(U_{ij}). \]

Local sections \(x_i\in X(U_i)\) are compatible when

\[ \rho ^i_{ij}(x_i)=\rho ^j_{ij}(x_j) \]

on every declared overlap. Predictive-state realizations are natural here because they describe a system through observable tests rather than a single assumed latent state [ Littman et al. , 2001 , Singh et al. , 2004b ] .

Define

\[ M_{\mathcal U}=\prod _iX(U_i), \qquad O_{\mathcal U}=\prod _{i,j}X(U_{ij}), \]

and let \(d_0,d_1:M_{\mathcal U}\rightrightarrows O_{\mathcal U}\) apply the two restrictions. The compatible-family object is

\[ Z_{\mathcal U}=\operatorname {Eq}(d_0,d_1). \]

A separate map

\[ \eta _{\mathcal U}:X(U)\longrightarrow Z_{\mathcal U} \]

restricts a declared global model to its local family.

Commutative diagram illustrating 18.1 Compatibility and effectivity.

Compatibility asks whether a family lies in \(Z_{\mathcal U}\). Effectivity asks whether it lies in the image of \(\eta _{\mathcal U}\). Local models can agree on every measured overlap while failing rank, normalization, causal, or dynamical constraints of the global model class.

Definition 18.1 SID factorization problem

The base SID obstruction is the obstruction to inhabiting both the compatibility and effectivity factorizations of the declared descent sketch. The tangent obstruction is the same factorization problem after applying the tangent functor to its objects, restrictions, and cones.

Design principle

Compatibility and effectivity are two typed obstructions, not two terms in one loss. They can demand different repairs: transport between local models, expansion of the global class, a new context, or refusal to promote.