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π1(S1)\pi_1(S^1)
≃\simeq
∑\sum
≡\equiv
→\to
Hong KongLaunching September 2026

HoTT.Cafe

Homotopy Type Theory Consultancy

(A≃B)≃(A=B)(A \simeq B) \simeq (A = B)

Expert guidance in formal verification, proof assistants, and the mathematics of type theory. Transform your approach to software correctness.

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What We Do

Our Services

From foundational research to practical implementation, we bring the power of type theory to your organization.

Formal Verification

Prove your software correct with mathematical certainty using dependent types and proof assistants.

⊢P:Correct(prog)\vdash P : \text{Correct}(\text{prog})

Type Theory Research

Advance the frontiers of HoTT, cubical type theory, and higher inductive types.

ua:(A≃B)→(A=B)\mathsf{ua} : (A \simeq B) \to (A = B)

Proof Assistant Training

Master Coq, Agda, Lean, or Idris with tailored training programs.

Γ⊢t:T\Gamma \vdash t : T

Mathematical Consulting

Apply category theory, homotopy theory, and algebraic structures to real-world challenges.

HomC(F(A),B)≅HomD(A,G(B))\text{Hom}_C(F(A), B) \cong \text{Hom}_D(A, G(B))

Code Audits

Deep analysis of critical systems using formal reasoning to identify logical flaws.

{P}  C  {Q}\{P\} \; C \; \{Q\}

Specification Design

Precise, unambiguous specifications that define exactly what your system should do.

Spec:U→Prop\text{Spec} : \mathcal{U} \to \text{Prop}
What We Offer

Where Theory Meets Practice

Deep mathematical expertise, applied to your unique challenges.

Research

Mathematical Research Partnership

Partner with us on cutting-edge HoTT research. We bring deep expertise in homotopy theory, higher category theory, and type-theoretic foundations to collaborative projects.

ua:(A≃B)→(A=UB)\mathsf{ua} : (A \simeq B) \to (A =_{\mathcal{U}} B)
Education

Tailored Training Programs

Master the art of formal reasoning. From introductory type theory to advanced HoTT seminars, we design learning experiences that transform how your team thinks about mathematics.

Γ⊢t:T\Gamma \vdash t : T
Tech

Proof Assistant Development

Leverage Coq, Agda, Lean, or Idris for your critical systems. We help you build verified software with mathematical certainty.

⊢P:Correct(prog)\vdash P : \text{Correct}(\text{prog})
Consulting

Strategic Advisory

Navigate complex mathematical challenges with expert guidance. We provide high-level consulting on foundations, methodology, and the application of abstract mathematics to real problems.

HomC(F(A),B)≅HomD(A,G(B))\text{Hom}_C(F(A), B) \cong \text{Hom}_D(A, G(B))
Tech

Formal Specification Design

Define precisely what your systems should do. We craft unambiguous specifications that serve as the foundation for verified implementations.

{P}  C  {Q}\{P\} \; C \; \{Q\}
Launching September 2026

Ready to Explore?

Join us on a journey through the landscape of Homotopy Type Theory. Let's discover what we can build together.

Get in Touch

Based in Hong Kong🌏Serving clients worldwide