ora-0167
15.1 The decision quotient
Let \(\mathbf{Pres}_{\mathcal U}\) be the category of decision presentations valid on \(\mathcal U\), with morphisms given by semantics-preserving translations. Localizing at the translations that intertwine realization, readout, and observer gives
Objects of \(\mathbf{Dec}_{\mathcal U}\) are decision mechanisms rather than surface algorithms. The subscript is indispensable: enlarging \(\mathcal U\) can invalidate an equivalence that held in the unconstrained Euclidean case.
Suppose the UDL semantics \(\mathsf U=\operatorname {Ran}_K\operatorname {Lan}_J\), its decision readout, and its observer send every arrow in \(W_{\mathcal U}\) to an equivalence. Then they factor essentially uniquely through \(\mathbf{Dec}_{\mathcal U}\). Any theorem expressed solely in those semantics is presentation invariant on \(\mathcal U\).
This is the universal property of localization. A functor that inverts \(W_{\mathcal U}\) factors through \(\mathbf{Pres}_{\mathcal U}[W_{\mathcal U}^{-1}]\), uniquely up to coherent equivalence. The same factorization for readout and observer makes every statement built from their images insensitive to the chosen representative.