Symbol Reference: Emb / ↪ — Canonical Embedding Operator (Deep-to-Shallow / Subsort / Universe Embedding)

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Formula Display:
Proof Trees:

Emb / ↪ \(\mathsf{Emb} \; / \; \hookrightarrow\) — Canonical Embedding Operator (Deep-to-Shallow / Subsort / Universe Embedding)

Canonically embeds a semantic class or lower-level object into the syntactic expression layer (numeral/constant injection) or embeds a sub-conglomerate into a larger domain.

Field Mathematical & Kernel Specification (MKTD1 / MKTD2)
Symbol
Emb / ↪
\(\mathsf{Emb} \; / \; \hookrightarrow\)
Category Embedding & Interpretation
SML AST Constructor Kernel AST Symbol / Sort Constructor in typetheory.ML (Class_p / Truth_p / EXP / UCol)
Constructor Arguments Typed arguments as specified by the Signature / Sort field above
Metatype / Signature Class → Expression | Sub-Universe ↪ Ambient Class
Binder Status See Category & Signature above
Defining Kernel Entries (ML Labels) See Related Symbols & Kernel Entries below.
LaTeX Form \mathsf{Emb} \; / \; \hookrightarrow
How to Understand It

How to Understand Embedding (Emb / ↪) in MKTD1 & MKTD2:

Whereas Enc : Expression → Class maps a syntactic expression to its code, Embedding Emb : Class → Expression (written symbolically with the hook arrow ↪) maps an object-level class X : Class to the canonical syntactic expression denoting X (analogous to Feferman's dot/numeral operator x ↦ ‹x› in modal/provability logic):

  • Semantic Evaluation Law: Evaluating the embedded expression Emb(X) under any valuation A returns the class X itself: μ_A(Emb(X)) = X.
  • Substitution into Quoted Formulas: Allows parameterizing quoted formulas by actual classes via ‹ 𝛂(⟨Emb(X)⟩) ›.
  • Variant 1 ↪ Variant 2 Bridge: Also denotes the structure-preserving embedding of Variant 1 first-order derivations into Variant 2 higher-order proof terms.
Canonical Law / Usage ∀(X:Class). (μ_A(Emb(X)) = X)
Related Symbols & Kernel Entries