Generic τ-object
A generic τ-object is an inhabitant of the τ-categorical kernel — an object of the (∞, 1)-category τ that arises from finite K1–K5 composition (closed under K6). Each τ-object carries a finite K1-trace identifying its construction; equivalence of τ-objects is decided by trace-equality, making the framework's object-equality a finite, decidable predicate.
τ-Definition
A generic τ-object is an inhabitant of the τ-categorical kernel — an object of the (∞, 1)-category τ that arises from finite K1–K5 composition (closed under K6). Each τ-object carries a finite K1-trace identifying its construction; equivalence of τ-objects is decided by trace-equality, making the framework's object-equality a finite, decidable predicate.
Categorical invariant. An object A ∈ τ — equivalently, a finite K1–K5-trace closed under K6.
Primary registry anchor:
I.D56
τ-Derivation Chain
-
I.K0— Universe Postulate — kernel τ exists -
I.K1— K1–K5 generators give the construction primitives -
I.K6— K6 closure restricts to finite K1–K5 composition -
I.D56— τ-site — τ as a category, whose objects are the inhabitants of τ
Lean modules referenced:
TauLib.BookI.Topos.EarnedTopos,
TauLib.BookI.Topos.LimitsSites
Mathematical content
A τ-object is an element of the object-class of the (∞, 1)-category τ — equivalently, a finite K1–K5-trace closed under K6.
Trace representation. Each τ-object A admits a canonical K1-trace: a finite sequence of K1–K5 generators whose composition produces A. Two τ-objects are equal iff their canonical traces are equal — making τ-object equality a decidable predicate.
Consequence. The τ-framework's object-equality is decidable. Combined with the Hyperfactorization theorem (T01) and the Categoricity theorem (T03), this means τ-objects can be enumerated, compared, and checked computationally — which is what allows the central theorem at rank (3, 15) (T04) to be a `native_decide` finite check rather than a meta-theoretic argument.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Topos.EarnedTopos
Lean kind: structure
Lean symbol: Tau.BookI.Topos.tauObject
Cross-domain bridges
This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.