Riehl Post-Rigorous Skill

Emily Riehl's post-rigorous mathematics framework: formalizing arguments that are "locally may contain errors but globally sound."

Trit: 0 (ERGODIC) — Coordinator between rigorous and synthetic approaches

Core Thesis

"We thought we were post-rigorous but it may have turned out we also have been pre-rigorous as well, oy vey!"

The Triad of Mathematical Maturity (Tao)

PRE-RIGOROUS → RIGOROUS → POST-RIGOROUS
     ↓              ↓            ↓
  intuitive      formal      synthetic
  examples       proofs      sketches
  hand-waving    details     globally-sound

Key Insight: Post-rigorous mathematics allows "local errors that cancel globally" — this is GF(3) conservation:

(-1) + 0 + (+1) = 0

Primary Sources

Papers

Talks

Three Formalization Strategies

Strategy 1: CONCRETE (Analytic)

Formalize in specific model (quasi-categories in Lean)
└─ Labor-intensive but complete
└─ Example: mathlib4 category theory
└─ Pros: Fully rigorous, computable
└─ Cons: Model-specific, verbose

Strategy 2: AXIOMATIC (∞-Cosmos)

Define ∞-cosmos axiomatically
└─ Prove theorems abstractly
└─ Show quasi-categories form ∞-cosmos
└─ Example: Elements of ∞-Category Theory
└─ Pros: Model-independent, elegant
└─ Cons: Requires axiom verification per model

Strategy 3: SYNTHETIC (Domain-Specific Type Theory)

Generalized elements = terms in context
└─ Informal proof IS formal proof
└─ Example: Rzk proof assistant
└─ THIS IS THE POST-RIGOROUS APPROACH
└─ Pros: Natural, directly formalizable
└─ Cons: New foundations, learning curve

Key Definitions (MathPix Extracted)

∞-Cosmos (Definition 1.2.1)

An ∞-cosmos $\mathcal{K}$ is a simplicially enriched category with:

$$\mathcal{K} \text{ has } \begin{cases} \text{(a) isofibrations closed under:} & \text{composition, pullback, cotensor} \ \text{(b) limits of towers:} & A_0 \twoheadleftarrow A_1 \twoheadleftarrow A_2 \twoheadleftarrow \cdots \ \text{(c) cotensors:} & A^X \text{ for } X \in \mathsf{sSet} \end{cases}$$

GF(3) Assignment:

• Isofibration: trit = -1 (constraint/validation)
• Equivalence: trit = 0 (transport)
• Trivial fibration: trit = +1 (generation)

Homotopy 2-Category (Definition 1.3.1)

The homotopy 2-category $\mathfrak{h}\mathcal{K}$ has:

• Objects: ∞-categories in $\mathcal{K}$
• 1-cells: ∞-functors
• 2-cells: Homotopy classes of natural transformations

$$\mathfrak{h}\mathcal{K}(A,B) := \pi_0(\mathrm{Fun}(A,B))$$

Adjunction in ∞-Cosmos (Proposition 1.1.3)

An adjunction $f \dashv u$ in $\mathfrak{h}\mathcal{K}$ consists of:

$$\begin{CD} A @>f>> B \ @. @VuVV \ @. A \end{CD} \quad \text{with } \eta: \mathrm{id}_A \Rightarrow uf, \quad \epsilon: fu \Rightarrow \mathrm{id}_B$$

satisfying triangle identities up to homotopy.

Representable Terminal Element (Proposition 2.1.2)

An element $t: \mathbf{1} \to A$ is terminal iff: $$\forall f: X \to A, \exists! \alpha: f \Rightarrow t \circ !$$

Diagrammatically:

     X
    f│   ⇓∃!
     ▼
1 ──t──▶ A

Comma ∞-Category (Definition 3.1.1)

Given $f: A \to C$ and $g: B \to C$, the comma ∞-category $f \downarrow g$ satisfies:

$$\begin{CD} f \downarrow g @>p_0>> A \ @Vp_1VV @VfVV \ B @>g>> C \end{CD}$$

with a 2-cell $\phi: f p_0 \Rightarrow g p_1$ universal among such squares.

Model Independence Theorem

Theorem 3.6: Categorical properties encoded by fibered equivalences between comma ∞-categories are:

1. Preserved by cosmological functors
2. Reflected by cosmological biequivalences

This includes:

• Adjunctions
• Limits/colimits
• Kan extensions
• Fibrations

Proof Assistants

Lean 4 (infinity-cosmos)

-- From plurigrid/infinity-cosmos
import InfinityCosmos.Basic

structure InfinityCosmos where
  carrier : Type*
  hom : carrier → carrier → QCat
  isofib : ∀ {A B}, (A ⟶ B) → Prop
  -- Axioms...

theorem model_independence (K₁ K₂ : InfinityCosmos) 
  (F : CosmologicalBiequiv K₁ K₂) :
  ∀ P : CategoricalProperty, P.holds_in K₁ ↔ P.holds_in K₂ := by
  sorry -- Main theorem

Rzk (Synthetic)

-- Synthetic ∞-category theory
#def is-segal (A : U) : U
  := (x y z : A) → (f : hom A x y) → (g : hom A y z)
     → is-contr (Σ (h : hom A x z), hom2 A x y z f g h)

#def yoneda-lemma (A : U) (is-segal-A : is-segal A)
  (a : A) (F : A → U) (is-covariant-F : is-covariant A F) :
  Equiv (F a) ((x : A) → hom A a x → F x)
  := ...

GF(3) Conservation in Post-Rigorous

Strategy 1 (Concrete):  trit = -1  (rigorous validation)
Strategy 2 (Axiomatic): trit = 0   (model-independent transport)
Strategy 3 (Synthetic): trit = +1  (generative synthesis)

Conservation: (-1) + 0 + (+1) = 0 ✓

Kapranov-Voevodsky Error Example

Historical: Claimed proof of a key lemma was later found incorrect.

Post-rigorous insight: The error was "local" — the global structure survived, and the gap was eventually filled by others.

Skill analog: If

skill-dispatch

dynamic-sufficiency

Hatchery Integration

plurigrid/infinity-cosmos

TeglonLabs/narya

emilyriehl/yoneda

rzk-lang/rzk

World Connections

MathPix Pipeline

# Extract diagrams from Riehl papers
mathpix convert_document \
  --path "elements-infinity-category-theory.pdf" \
  --formats latex \
  --checkpoint-seed 1069

# Convert to ACSet schema
julia -e '
using MathpixACSet
acset = pdf_to_acset("riehl-verity-scratch.pdf",
  seed=0x42D,
  checkpoint_pattern=SEED_1069)
'

Key Diagram Extraction

% Homotopy 2-category
\begin{tikzcd}
A \ar[r, "f"] \ar[d, "g"'] & B \ar[d, "h"] \\
C \ar[r, "k"'] & D
\ar[from=1-1, to=2-2, Rightarrow, shorten <=3pt, shorten >=3pt, "\alpha"]
\end{tikzcd}

% Comma category universal property
\begin{tikzcd}
f \downarrow g \ar[r, "p_0"] \ar[d, "p_1"'] & A \ar[d, "f"] \\
B \ar[r, "g"'] & C
\ar[from=1-1, to=2-2, Rightarrow, shorten <=5pt, shorten >=5pt, "\phi"]
\end{tikzcd}

Triadic Skill Integration

elements-infinity-cats (-1) ⊗ riehl-post-rigorous (0) ⊗ infinity-topos (+1) = 0 ✓

elements-infinity-cats

riehl-post-rigorous

infinity-topos

Commands

# Fetch latest slides
curl -O https://emilyriehl.github.io/files/post-rigorous.pdf

# Convert to LaTeX
mathpix smart_pdf_batch --path post-rigorous.pdf

# Verify in Rzk
rzk typecheck synthetic-proof.rzk

# Check in Lean
lake build InfinityCosmos

References

• Riehl, E. (2024). "Formalizing post-rigorous mathematics." Hausdorff Center.
• Riehl, E. (2023). "Could ∞-category theory be taught to undergraduates?" arXiv:2302.07855.
• Riehl, E. & Verity, D. (2020). "∞-category theory from scratch." Higher Structures 4(1).
• Riehl, E. & Verity, D. (2022). Elements of ∞-Category Theory. Cambridge.
• Riehl, E. & Shulman, M. (2017). "A type theory for synthetic ∞-categories." Higher Structures 1(1).
• Tao, T. (2007). "There's more to mathematics than rigour and proofs."

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.