emdash — Functorial Type Theory

emdash is a research programme and executable formalization for functorial type theory: categorical action is part of the computational language rather than an external structure added after the fact. The development combines dependent type theory with categories, directed families, functors, transfors, and higher cells, using cut-elimination-inspired operations so that functoriality and naturality can compute.

The current v3.2 edition is a checked development draft and a working, bounded product—not a finished foundation, complete proof assistant, or claim of global metatheory.

Read And Review

Start with the concise Functorial Type Theory: An Executable Architecture for Directed Dependency overview, then continue to the current development edition of Functorial Type Theory: Univalent Foundations for Mathematics (assembled Markdown). The active mathematical source is emdash2/emdash3_2.lp, together with the modules it imports.

Use the live integrated reviewer to elaborate the bounded categorical syntax, inspect explicit Core and computation, run the three-panel research report, and read the book in the same client-side workbench.

To run that reviewer locally after bootstrapping a fresh checkout:

./scripts/bootstrap-worktree.sh
./scripts/pnpmw run reviewer:dev

For a compact terminal walkthrough of the same architecture:

./scripts/pnpmw run demo:external-review

The browser workbench is published from main by GitHub Pages. It is wholly client-side and does not require a Lambdapi process or other production backend.

What The Reviewer Shows

The reviewer brings four parts of the project into one place:

• an outer dependent logical framework, including a Sigma-telescope example;
• ordinary, natural, displayed-functorial, and displayed-natural categorical binders in their reviewed profiles;
• readable source, backend-neutral explicit emdash Core, inferred types, structural lowering, computation, and source-located failures; and
• the mathematical book alongside the executable examples and preserved minimal-Core playground.

The examples include nested functorial abstraction and a genuinely dependent displayed chain. They are intended to expose the architecture and its present boundary, not to simulate completion of the book's entire notation.

Architecture And Authority

Layer	Present role
Active Lambdapi v3.2 development	Authors and checks the categorical declarations, computation rules, and proof-time comparisons. It remains the mathematical authority.
TypeScript surface and explicit Core	Recursively elaborates a reviewed direct-TypeScript and textual surface into backend-neutral explicit owners.
Generic TypeScript dependent LF	Checks explicit Core, performs conversion and bounded reduction, and runs entirely in the client for the reviewer profile.
Lambdapi conformance route	Optionally emits deterministic judgments and compares selected results with the active kernel. It is a development oracle, not a production backend.

The TypeScript implementation is therefore a real small checker/evaluator, but only for its recorded profile. Readable syntax may omit parameters that bidirectional typing can recover; it may not invent categorical action or external naturality evidence when no internal construction owns it.

Current Boundaries

• The text adapter is not a parser for every notation in the book or for arbitrary Lambdapi source. It accepts the reviewed mathematical constructions and fails closed outside them.
• Displayed contexts demonstrate independent fibrewise siblings, genuine dependency, a canonical mixed telescope, and qualified depth-generic finite Hom-category recursion. Arbitrary variance DAGs, general mixed introduction/curry, exchange across dependency, and unrestricted displayed coherence remain open.
• The remaining Lambdapi library has not been proven mechanically transferable as one batch. Bulk transfer qualification is deliberately deferred to a future goal.
• Groupoidal specialization/closure and general normalization, confluence, canonicity, consistency, and semantic soundness for the combined calculus are not claimed.

These are continuation boundaries, not hidden assumptions of the examples that already run.

Contributor Setup And Focused Commands

Node 22.13 or newer is required. The repository uses the pinned pnpm wrapper and one workspace lockfile; Lambdapi is additionally required for formal kernel and conformance checks.

./scripts/bootstrap-worktree.sh

Use the smallest gate that covers a change:

./scripts/pnpmw run reviewer:dev        # integrated local browser reviewer
./scripts/pnpmw run demo:external-review
./scripts/pnpmw run check:ts            # root TypeScript workbench
./scripts/pnpmw run kernel:check         # active Lambdapi kernel
./scripts/pnpmw run book:check           # authored book contracts

Repository workflow and authority are defined in AGENTS.md; formal-kernel changes also follow emdash2/AGENTS.md. Renewed TypeScript work starts from the v3.2 elaborator handoff, which routes to the living plans and detailed validation history.

Related Projects

• Arrowgram — diagrams and structured technical documents.
• Hotdocx — browser publishing and research workspaces.
• LastRevision.pro — hosted AI workspaces and automation.

Historical TypeScript Prototype

Repository history and parts of the root workbench preserve an earlier dependent-language feasibility prototype with bidirectional elaboration, holes, unification, rewriting, and proof-state machinery. Those generic mechanisms remain useful evidence, but the prototype's old category-specific design is not an authority for v3.2. The elaborator handoff records what was retained, replaced, and graduated in the renewed architecture.