Hi ¥Yuan, Your ProofScaffold you are describing is essentially an instance of “MathOps”!
TL;DR: I have developed the `arrowgram` commutative-diagrams editor https://hotdocx.github.io/arrowgram then I setup GPT-5.3-Codex inside my AI workspace to read the JSON text of those diagrams and generate the `emdash` proof assistant for lax ω-categories https://github.com/hotdocx/emdash (and to generate its report https://hotdocx.github.io/r/26043CPAL64001 as a book/slides with embedeed `arrowgram` diagrams), and you can try (and clone) my shared AI workspace and fund it at GetPaidX.com a.k.a. https://LastRevision.pro/r/26044DLGJ77000 — and probably, you should disregard the rest of this email. Bye. --- APPENDIX: What is MathOps? Indeed, as soon as you can setup some « MathOps » (i.e. math DevOps engineering) for a long-running “LLM ↔ proof-checker” feedback loop then autoformalization (e.g. “≈130k lines in two weeks”) is expected (strangely, Terminal-Bench 2.0 has no such benchmarks!). For category theory, the “proof-checker” (i.e. computational logic) question has been solved since Kosta Dosen “Cut-Elimination in Categories” (1999), but the tools to easily specify/implement it were not available until Frédéric Blanqui's Lambdapi logical framework (rewrite/unification rules) and: emdash 2 — Functorial programming for strict/lax ω-categories in Lambdapi https://github.com/1337777/cartier/blob/master/cartierSolution19.lp (99% generated by AI, lol) https://hotdocx.github.io/r/26043CPAL64001 And with good “mathops/devops engineering”, a general LLM such as GPT-5.3-Codex with Codex CLI has succeeded in coding this whole ≈4k lines file `cartierSolution19.lp` from just text prompts containing `arrowgram` commutative diagrams which hinted at this “dependent hom/arrow/comma category” construction: ## 1) Dependent arrow/comma/hom for a dependent category and “stacking” pasting diagrams We use displayed/dependent categories over a base category `B` (i.e. a fibration `E : B → Cat`, or a more general isofibration `E → B` without transport/arrow-action). For `b₀ ∈ B` and `e₀ ∈ E(b₀)`, the dependent hom/comma/arrow-category is the functor: - `Homd_E(e₀, (–,–)) : E ×_B (Hom_B(b₀, –))ᵒᵖ → Cat` - notations: `Homd_E : Π b₀, E[b₀]ᵒᵖ → (Σ b₁, E[b₁] × Hom_B(b₀, b₁)ᵒᵖ) → Cat` On a fibre object `e₁ ∈ E(b₁)` and a base arrow `b₀₁ : b₀ → b₁`, it returns the fibre hom-category, where `(b₀₁)! e₀` is the `E`-action/transport of `e₀` along `b₀₁`: - `Homd_E(e₀, (e₁ , b₀₁)) ≔ Hom_{E(b₁)}( (b₀₁)! e₀ , e₁ )` That is, this syntactic triangle/simplicial classifier `homd_`, reduces to `globular` semantics, but is necessary for iterating simplicially (triangles → tetrahedra → …) and for expressing non-cartesian lax ω-functors and lax ω-transfors. Moreover the functoriality of `Homd_E(e₀, (–,–))`, especially in its second argument `(Hom_B(b₀, –))ᵒᵖ`, espresses precisely the “stacking” of 2-cells along a 1-cell (i.e. generalization of horizontal composition of 2-cells meeting at a 0-cell). https://hotdocx.github.io/r/26043CPAL64002 This dependent comma/arrow/hom category `homd_` is analoguous to the (directed) “bridge type” construction in the technique of logical relations/parametricity, i.e. Ambrus Kaposi “Internal parametricity, without an interval”; and it is really an *internalized* type-theory/computational-logic as contrasted from Hugo Herbelin “A parametricity-based formalization of semi-simplicial and semi-cubical sets” which is merely a (formalized) combinatorial analysis. ## 2) Internalized computational-logic and “syntax elaboration” The `emdash` Lambdapi specification looks like the categorical semantics (“categories-with-families”) of a programming language; but because it is carefully formulated to be “computational” and its dependent-types/logic is “internalized”, then it can actually be translated as a traditional programming-language/proof-assistant surface syntax ( https://github.com/hotdocx/emdash ) whose HOAS bidirectional type-checking and elaboration/engineering in TypeScript is easily guaranteed. For example, the fibred/displayed functor `FF : Functord_(Z) E D` between isofibrations `E D : Catd Z` over `Z` is read as `z :^f Z, e :^f E[z] ⊢ FF[e] : D[z]` (where binders `:^f` are functorial and `:^n` are natural, and `:^o` is binding over objects-only). Thus `Functord` is the traditional Π-type (`Functord_(Z) E D ≡ Π_(Z) (Functor_catd E D)`), but with a construction `fapp1_funcd`, to express *lax* functoriality in the `e : E[z]` variable (where `→_` is dependent hom `homd_`, is used to express that this expression may be non-cartesian in general, i.e. *lax* functor): - `z :^f Z, e :^n E[z], z' :^f Z, f :^n z → z', e' :^n E[z'], σ :^f e →_f e' ⊢ FF₁(σ) : FF[e] →_f FF[e']` Similarly from the usual “diagonal” components `z :^f Z, e :^n E[z] ⊢ ϵ[e] : FF[e] → GG[e]` for a displayed transfor/transformation `ϵ : Transfd_(Z) FF GG`, there is a construction `tapp1_fapp0` for “off-diagonal” components (i.e. the index/subscript is an arrow instead of an object), to express *lax* naturality/functoriality in the `e : E[z]` variable: - `ϵ :^f Transfd FF GG, z :^f Z, e :^n E[z], z' :^f Z, f :^n z → z', e' :^n E[z'], σ :^f e →_f e' ⊢ ϵ_(σ) : FF[e] →_f GG[e']` These constructions are expressed *internally* (as stable head symbols `fdapp1_int_transfd` and `tdapp1_int_func_transfd`, etc... the discharge/abstraction/lambda of the full (or partial) list of variables in the context-patterns above), therefore those newly introduced variables vary functorially/naturally. Note that when `E : Functor Z Cat` is a module/profunctor, then the exchange of the order of binder variables (i.e. “module-transport” vs “functor-action”) must be an explicit operation: - `z :^n Z, e :^f E[z], z' :^n Z, f :^f z → z' ⊢ (f)_!(e) ≔ E₁(f)(e) : E[z']` And because there is available a “context-extension” / total-category / Sigma-category construction `Total_cat E : Cat` (a.k.a. `Σ_(B) E : Cat`) for any fibred category `E : Catd B`, all these surface syntax can actually happen within any ambient context `Γ, ⋯ ⊢` (i.e. the base `Z` is itself `Total_cat Z0` for `Z0 : Catd Γ`). And the usual categorical-logic fact that the Sigma/total-category `Σ b, –` is left-adjoint to weakening/lift `–[b]`, means that the dependent hom/arrow-category has an alternative formulation (`homd_curry` in the lambdapi spec file `cartierSolution19.lp`): - `Homd_E : Π b₀, E[b₀]ᵒᵖ → Π b₁, Hom_B(b₀, b₁)ᵒᵖ → E[b₁] → Cat[b₁]` In reality, the *internal* computational-logic for lax ω-categories is *easier* to express than for only strict 1-categories; because the hom/comma of a category `Hom_D(y, F –)` (i.e. comma vs hom, is necessary) is recursively a (fibred) category, and the arrow-action of a *lax* functor `F₁ : Hom_C(x, –) → Hom_D(F₀ x, F –)` is recursively a fibred functor which is non-cartesian (because of laxness), i.e. a commuting-triangle/identity 2-arrow is mapped to a non-identity 2-arrow by `((F₁)₁)₀ : Homd_(Hom_C(x, –))(f, (g ∘ f, g)) → Homd_(_)((F₁)₀ f, ((F₁)₀ (g ∘ f), (F₁)₀ g))`. But for ω-iteration of the natural n-transfors construction (i.e natural family rather than an individual n-arrow), there needs to be a way to re-interpret a natural transformation again as a functor: this is the question solved by the next section. ## 3) Naturality as “cut accumulation” and the exchange law A (fibred) transformation: `F ⇒ G`, is also a transformation from the unit/hom profunctor (i.e. “off-diagonal”): `hom(~, —) ⇒ hom(F ~, G —)`, and is back again as a fibred functor between isofibrations: `∫ hom(~, —) → ∫ hom(F ~, G —)`; thus we can continue the ω-iteration of this construction. The key point is that *naturality can be oriented as an “accumulation” rewrite* on these transfor's “off-diagonal” components `ϵ_(–)` (cut-elimination style, where `⋅` is the vertical compositon/cut): - `(G b) ⋅ ϵ_(a) ↪ ϵ_(b⋅a)` - `ϵ_(b) ⋅ (F a) ↪ ϵ_(b⋅a)` An instance of this accumulation rule is the *exchange law* between horizontal and vertical compositions: an unambiguous “pasting diagram” with two vertical 2-cells `α ≔ a` then `β ≔ b`, and one horizontal 2-cell `ϵ ≔ (e ∘ —)` for `e : f → g` (where `G ≔ (g ∘ —)` and `F ≔ (f ∘ —)` are horizontal post-composition/whiskering), will normalize to a unique form (where `∘` is horizontal composition): - `(g ∘ β) ⋅ (e ∘ α) ↪ e ∘ (β⋅α)` https://hotdocx.github.io/r/26043CPAL65001 Moreover such formulations of naturality are useful to express computationally the triangular identities of *adjoint functors*: - `ϵ_(f) ∘ LAdj(η_(g)) ↪ f ∘ LAdj(g)` and for sheaves and schemes, polynomial functors, higher inductive types, etc... (ref: `cartierSolution16.lp`) ## 4) GPT-5.3-Codex diagrammatic prompting via `arrowgram` and “MathOps” What is MathOps? Example: How to enable the GPT-5.3-Codex + Codex CLI coding agent to (natively) understand and generate commutative diagrams? A MathOps solution is: `arrowgram` https://github.com/hotdocx/arrowgram/ an open-source strict JSON text specification for commutative diagrams with exporting to JSON/SVG/PNG/TikZ and with an AI-editor that generates and renders `markdown` papers documents and slides presentations with embedded diagrams specs, from any uploaded source references. While the core `arrowgram` open-source app can be downloaded and used offline, there is an academic-publishing overlay online SaaS app: https://hotdocx.github.io to enable rapid/iterative MathOps from idea, to sharing, to community peer review! MathOps is what happens after you get “desk-rejected” by POPL 2026, lol... A MathOps solution is the ability to share workspace sessions (docker), with pre-installed running AI coding agents + proof-checkers, for “vibe coding” live with co-workers, in a marketplace where large-scale community-workspaces can get paid and funded by fans and local businesses and governments; bypassing well-known falsifications and intoxications. This MathOps idea is implemented by https://GetPaidX.com — Publish and fund AI workspaces backed by AI communities of 20,000+ developers in Dubai, Mumbai, Shanghai https://meetup.com/dubai-ai https://meetup.com/mumbai-ai https://meetup.com/shanghai-ai ... Try and experiment the AI workspace for `emdash` here (e.g ask Codex “where is the dependent hom defined? is it correct”): https://GetPaidX.com/r/26044DLGJ77000 --- References: [1] Kosta Dosen, Zoran Petric (1999). “Cut-Elimination in Categories” [2] This summer visit to Ambrus Kaposi in Budapest, and `emdash` discussions at RHPL@FSTTCS 2025 in India [3] GPT-5.3-Codex -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/metamath/CAOrfhcFdp46qZ3Cr7mpQ0Z%2BHh6cc%2B9s_%2BDV43UfHs4ZdfuVPRg%40mail.gmail.com.
