limits-colimits

# 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 ```bash # Lean 4: import CategoryTheory.Limits.Shapes.Products ``` ### Lean4_Universal ```bash # Lean 4: IsLimit.lift cone -- unique morphism from universal pr...