quotientMapLeft
plain-language theorem explainer
quotientMapLeft defines the canonical projection from the recognition quotient of a composite recognizer to the quotient of its first component. Researchers in recognition geometry cite this map when establishing that combined detectors produce finer partitions of configuration space. The definition is realized as a one-line wrapper that lifts the marking map through the left refinement condition.
Claim. Let $R_1$ and $R_2$ be recognizers on a configuration space $C$. The map $Q(R_1 ⊗ R_2) → Q(R_1)$ sends the equivalence class of a configuration under the composite indistinguishability relation to its class under the indistinguishability relation of $R_1$, where $Q$ denotes the recognition quotient and ⊗ the composite recognizer.
background
Recognition geometry examines how recognizers induce quotients on configuration spaces. A recognizer maps configurations to events and thereby defines an indistinguishability relation; the recognition quotient is the set of equivalence classes under that relation. The composite recognizer $R_1 ⊗ R_2$ distinguishes a pair of configurations precisely when at least one of the components distinguishes them, so its quotient is finer than either factor quotient. The module develops the theory of such composites and proves the refinement theorem asserting that the composite quotient maps onto each component quotient. This construction depends on the left refinement lemma composite_refines_left, which states that indistinguishability under the composite implies indistinguishability under the left factor, together with the equality condition quotientMk_eq_iff that characterizes when two markings determine the same class.
proof idea
The definition proceeds by Quotient.lift applied to the marking map recognitionQuotientMk r₁. Well-definedness on equivalence classes is verified by feeding the hypothesis of the lift into composite_refines_left r₁ r₂ and then applying (quotientMk_eq_iff r₁).mpr to obtain the required equality of classes in the target quotient.
why it matters
This definition supplies the left projection required by the refinement theorem, which asserts surjectivity of both component maps and is the central result of the composition module. It is invoked directly in refinement_theorem and in the paper-level statement refinement that records the surjectivity claim. Within the Recognition Science framework the map realizes the geometric enrichment produced by combining measurements, consistent with the composition law that underlies the emergence of richer structure from simpler recognizers.
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