6.5.5 Tensor Valence Type Relation

Tensor Valence Type Relation is the formal statement that the map sending a valence sequence to its underlying type is a well-defined, surjective, many-to-one function, associating to each valence sequence exactly one type but associating to each type, in general, several distinct valence sequences, with the fiber over a given type (p, q) containing exactly (p+q)!/(p!q!) valences. This relation makes precise, in the language of functions between sets, the informal statement that valence "refines" type: it exhibits type as a quotient of valence, obtained by forgetting positional information while retaining count information.

The Map From Valence to Type

Defining the Map

Let V_k denote the set of all valence sequences of length k, and let Types_k denote the set of all types (p, q) with p + q = k. The map

sends a valence sequence (σ₁, ..., σ_k) to the pair (p, q) obtained by counting its up-arrows and down-arrows, exactly as already established for the contravariant and covariant counts individually.

Well-Definedness

The map τ is well-defined because every valence sequence has a uniquely determined count of up-arrows and down-arrows; there is no ambiguity in which type a given valence maps to, since counting symbols in a fixed sequence is a deterministic operation.

Surjectivity

The map τ is surjective onto Types_k, since for any type (p, q) with p + q = k, the canonical sorted valence (↑, ..., ↑, ↓, ..., ↓) with p up-arrows followed by q down-arrows is a valid valence sequence mapping to exactly that type; every type is therefore hit by at least one valence.

The Fibers of the Map

Counting the Fiber Over a Fixed Type

The fiber τ⁻¹((p,q)), the set of all valences mapping to a given type, has size:

the number of interleavings of p up-arrows and q down-arrows, matching the binomial-coefficient count already established under the slot-distribution framing.

Non-Injectivity When Either p or q Exceeds One

The map τ fails to be injective whenever the fiber size exceeds 1, which happens exactly when both p ≥ 1 and q ≥ 1 with at least one of them at least 2 — for instance, type (2,1) has fiber size 3, meaning three genuinely different valences all map to the same type under τ.

Diagram of the Map as a Many-to-One Function

Type as a Quotient of Valence

The Equivalence Relation Induced by τ

The map τ induces an equivalence relation on V_k: two valences are equivalent exactly when they share the same up-arrow and down-arrow counts, that is, when one is a permutation of the other's positions; the set of equivalence classes under this relation is in bijection with Types_k, exhibiting type as the quotient set V_k / ~.

What Is Preserved and What Is Forgotten

Passing from a valence to its type preserves the two counts p and q exactly, but forgets everything about which specific positions carried which variance; this is the precise sense in which type is "coarser" than valence — it is valence modulo the forgetting of positional arrangement.

Consequences of the Relation for Operations

Operations Well-Defined on Type Must Respect the Fibers

An operation or property is well-defined as a function of type alone only if it takes the same value on every valence within a single fiber of τ; the total slot count k = p + q is one such property, since it depends only on the counts and not on arrangement, while "which slot is contracted first" is not, since different valences in the same fiber can disagree about which position is which.

Contraction Requires Lifting to a Specific Valence

Because contraction must specify particular slot positions, it cannot be described purely as an operation on types; it must be described as an operation on a chosen valence within a fiber, after which the type-level effect (subtracting (1,1) from (p,q)) is obtained by applying τ to the result.

Why This Relation Matters

Making Precise the Informal Notion of Refinement

Stating the valence-to-type relation as a specific surjective map with explicitly computable fiber sizes replaces the informal claim that "valence refines type" with a precise mathematical structure, allowing questions like "how much information does valence add beyond type" to be answered exactly, as the logarithm of the fiber size.

Clarifying When Type-Level Reasoning Suffices

Recognizing which properties factor through τ and which do not clarifies, before any specific calculation is attempted, whether working with type alone will suffice or whether a specific valence must be chosen first, preventing wasted effort applying type-level reasoning to a genuinely valence-dependent question.