Temporalization Operators

The temporalization operators (VII.D53) are the endofunctors T_past, T_present, T_future on the category Prop of propositions. Each operator relocates a proposition relative to the speaker's present, with three structural conditions: non-commutativity (past∘future ≠ present), idempotence failure (past∘past ≠ past, encoding pluperfect depth), and present-as-unit (T_present∘T_X = T_X). Tense, aspect, and mood are functors — not lexical tokens.

Categorical invariant. T_past, T_present, T_future : Prop → Prop with (i) non-commutativity T_past∘T_future ≠ T_present, (ii) idempotence failure T_past∘T_past ≠ T_past, (iii) T_present as left/right unit. The temporal structure of language is the algebra these endofunctors generate.

Primary registry anchor: VII.D53

Supporting items: VII.D51, VII.D45, VII.D54

1. I.K0 — Universe Postulate
2. VII.D45 — Temporal Experience Structure — pre-symbolic temporality
3. VII.D51 — Language Adds Temporalization — symbolic time emerges from language
4. VII.D53 — Temporalization Operators — tense/aspect/mood as endofunctors on Prop

Lean modules referenced: TauLib.BookVII.Meta.Saturation

The temporalization operators are instantiated whenever a speaker locates a proposition in time. Examples: simple past 'it happened' = T_past(p); pluperfect 'it had happened' = T_past(T_past(p)); future-in-the-past 'it was going to happen' = T_past(T_future(p)); present-as-unit 'it is the case that p' = T_present(p) = p.

Examples:

• Simple past — 'it happened': T_past(p), single past displacement
• Pluperfect — 'it had happened': T_past(T_past(p)), iterated past — depth, not collapse
• Future-in-the-past — 'it was going to happen': T_past(T_future(p)) ≠ T_present(p), composition records layering
• Present unit — 'it is the case that p': T_present(p) = p, the temporal zero adds no displacement

Register codomain: Prop (the category of propositions) within Reg_D (diagrammatic register) — temporalization is a structural feature of propositional content, not an empirical claim

Manuscript reference: manuscript-sources/book-07/part05/ch56.tex

Status: Formalized

Module: TauLib.BookVII.Meta.Saturation

Lean kind: structure

Lean symbol: Tau.BookVII.Meta.Saturation.TemporalizationOperators

Related glossary entries

• MG-A05-pragmatic-update-operator Pragmatic Update Operator
• MG-A03-tau-modal-operators τ-Modal Operators (□, ◇)
• MG-R03-diagrammatic-register Diagrammatic Register (Reg_D)

Referenced by

• MG-A05-pragmatic-update-operator Pragmatic Update Operator
• MG-A11-categorical-phenomenology-framework Categorical Phenomenology Framework