Intentionality (aboutness as morphism)

Intentionality is τ-categorical aboutness: the directional internal morphism in the Mind-Topos that takes a representation to its intentional object. It is not a mysterious extra relation linking mental content to the world — it is the structured arrow `Int : Rep → Obj` whose composition discipline (sense, reference, attitude) lives entirely within the agent's internal topos.

Categorical invariant. Intentionality = internal morphism Int: Rep → Obj in the Mind-Topos with directional structure and compositional closure (VII.D84). Brentano's mark of the mental is recovered as: every mental state is the codomain of an Int-morphism.

Primary registry anchor: VII.D84

Supporting items: VII.T39, VII.D82, VII.D44

1. I.K0 — Universe Postulate
2. VII.D82 — Mind as Internal Topos — carrier for representations and inferences
3. VII.T39 — Mind-Topos Structure Theorem — internal logic, beliefs as sections
4. VII.D84 — Intentionality as Morphism — aboutness with directional structure and compositional closure

Lean modules referenced: TauLib.BookVII.Logos.Sector

Intentionality is instantiated whenever a mental state is *about* something — a perception of an apple, a belief about Vienna, a desire for rest, a fear of falling. The aboutness is not added to the mental state from outside; it is the directional morphism that constitutes that state's content.

Examples:

• Belief that water is H₂O — Int-morphism from belief-token to chemical-structure object
• Desire for coffee — Int-morphism from desire-state to a not-yet-actualized object
• Fear of an unsigned letter — Int-morphism whose codomain is a fictional/non-existent object (still well-typed inside the topos)
• Sense vs reference (Frege): two distinct Int-morphisms with overlapping codomains — same object, different presentation arrows

Register codomain: Reg_E primarily (Int-morphisms ground belief content read as Obs), but every register can be the readout for an Int-image — Reg_P reads desires/intentions, Reg_C reads stance-directed acts.

Manuscript reference: manuscript-sources/book-07/part09/ch109.tex

Status: Formalized

Module: TauLib.BookVII.Logos.Sector

Lean kind: def

Lean symbol: Tau.BookVII.Logos.Sector.IntentionalityAsMorphism

Related glossary entries

• MG-H01-qualia Qualia (subsymbolic morphisms)
• MG-H03-mind-topos Mind-Topos Structure
• MG-R01-empirical-register Empirical Register (Reg_E)

Referenced by

• MG-H03-mind-topos Mind-Topos Structure