Bicategorical Yoneda Lemma in Lean 4

A Lean 4 / Mathlib formalization of the Yoneda lemma for bicategories.

Mathematical Statement

The classical Yoneda lemma identifies natural transformations out of a representable functor with elements of the represented set. In the bicategorical setting this bijection upgrades to a natural equivalence of categories: for a bicategory $\mathcal{B}$ and a pseudofunctor $F : \mathcal{B}^{\mathrm{op}} \to \mathbf{Cat}$, there is an equivalence

$$\mathbf{StrongTrans}(\mathcal{Y}_0(b),, F) ;\simeq; F(b)$$

natural in $b \in \mathcal{B}^{\mathrm{op}}$ and $F$, where $\mathcal{Y}_0(b)$ is the representable pseudofunctor sending $a \mapsto \mathcal{B}(a, b)$.

Rather than constructing this equivalence object-by-object, the formalization expresses it as a biequivalence of pseudofunctors

$$\mathtt{yonedaPairing} \simeq \mathtt{yonedaEvaluation}$$

on the product bicategory $\mathcal{B}^{\mathrm{op}} \times [\mathcal{B}^{\mathrm{op}}, \mathbf{Cat}]$.

Key Definitions

Lean name	Mathematical meaning
yonedaPairing	Pseudofunctor $(b, F) \mapsto \mathbf{StrongTrans}(\mathcal{Y}_0(b), F)$
yonedaEvaluation	Pseudofunctor $(b, F) \mapsto F(b)$
yonedaLemmaForwards	Strong transformation $\Phi : P \to E$, evaluating at $\mathrm{id}_b$
yonedaLemmaBackwards	Strong transformation $\Psi : E \to P$, sending $s$ to $(a, f) \mapsto F(f)(s)$
yonedaHomInvId	Unit isomorphism $\Psi \circ \Phi \cong \mathrm{id}_P$
yonedaInvHomId	Counit isomorphism $\Phi \circ \Psi \cong \mathrm{id}_E$
yonedaLemma	The assembled BiEquiv witnessing the equivalence
catPseudoULift	Universe-lifting pseudofunctor needed to type-match the two sides

Repository Structure

biyoneda/
├── Biyoneda/
│   ├── Basic.lean          # All definitions and proofs
│   └── ForMathlib.lean     # General Cat/bicategory lemmas staged for upstreaming
├── paper/
│   ├── Biyoneda.tex        # Paper source
│   ├── preamble.tex        # LaTeX packages and macros
│   └── references.bib      # Bibliography
├── Biyoneda.pdf            # Compiled paper (built by CI)
├── Makefile                # `make` to compile the paper locally
├── lakefile.toml
└── lean-toolchain          # Lean 4 v4.29.0 / Mathlib v4.29.0

Paper

A write-up of the formalization is in paper/Biyoneda.tex. The compiled PDF is kept up to date by CI and can be read directly on GitHub: Biyoneda.pdf.

To compile the paper locally (requires a TeX Live installation):

make

Building the Lean Code

Prerequisites: Lean 4 with elan (the Lean version manager). The correct toolchain is pinned in lean-toolchain and will be fetched automatically.

lake exe cache get   # download prebuilt Mathlib artifacts
lake build           # build the project

The first lake exe cache get step downloads compiled Mathlib oleans so you do not need to rebuild Mathlib from scratch (~20 min without cache).

Dependencies

• Lean 4 v4.29.0
• Mathlib4 v4.29.0

Specifically, the formalization builds on:

• Mathlib.CategoryTheory.Bicategory.Yoneda
• Mathlib.CategoryTheory.Category.ULift

Notes on AI

I have been experimenting with use of AI in formalization. I have had some limited success with it writing the lean code succses but I have used it for comments, documentation, and setting up the git actions in the repo. I have given a cursory look at all of the things it has written and corrected any small errors I have found. I plan to give a more detailed look at it in the future. Please be warned to take these things with a grain of salt.

License

Released under the Apache License 2.0, matching the Mathlib ecosystem. See LICENSE.

References

• J. Bénabou, Introduction to bicategories, 1967.
• N. Johnson and D. Yau, 2-Dimensional Categories, Oxford University Press, 2021. (arXiv)
• T. Leinster, Higher Operads, Higher Categories, Cambridge University Press, 2004. (arXiv)
• The Mathlib Community, Mathlib4, 2024. (GitHub)