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

2. Verify Universal Property

   • Cone from L with projections pi_i: L -> D_i
   • For any cone from X, unique morphism u: X -> L
   • Triangles commute: pi_i . u = cone_i
   • Lean 4:

     IsLimit.lift

     gives the unique morphism

3. Colimit (Dual)

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

4. 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)"

5. 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.