Limits Colimits

When to Use

Use this skill when working on limits-colimits problems in category theory.

Decision Tree

1. Identify Limit Type

- Product: limit of discrete diagram - Equalizer: limit of parallel pair f, g: A -> B - Pullback: limit of A -> C <- B - Terminal object: limit of empty diagram - Lean 4: CategoryTheory.Limits namespace

1. Verify Universal Property

- Cone from L with projections pii: L -> Di - For any cone from X, unique morphism u: X -> L - Triangles commute: pii . u = conei - Lean 4: IsLimit.lift gives the unique morphism

1. Colimit (Dual)

- Coproduct: colimit of discrete diagram - Coequalizer: colimit of parallel pair - Pushout: colimit of A <- C -> B - Initial object: colimit of empty diagram

1. Compute Limits Concretely

- In Set: product = Cartesian product - Equalizer = {x | f(x) = g(x)} - Pullback = {(a,b) | f(a) = g(b)} - sympy_compute.py solve "f(a) == g(b)"

1. Preservation

- Right adjoint preserves limits - Left adjoint preserves colimits - Representable functors preserve limits - Lean 4: Adjunction.rightAdjointPreservesLimits - See: .claude/skills/lean4-limits/SKILL.md for exact syntax

Tool Commands

Lean4_Limit

# Lean 4: import CategoryTheory.Limits.Shapes.Products

Lean4_Universal

# Lean 4: IsLimit.lift cone -- unique morphism from universal property

Sympy_Pullback

uv run python -m runtime.harness scripts/sympy_compute.py solve "f(a) == g(b)"

Lean4_Build

lake build  # Compiler-in-the-loop verification

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.