limits-colimits
Provides strategies for computing limits and colimits in category theory.
Install
mkdir -p .claude/skills/limits-colimits && curl -L -o skill.zip "https://agentskills.codes/api/skills/download/5283" && unzip -o skill.zip -d .claude/skills/limits-colimits && rm skill.zipInstalls to .claude/skills/limits-colimits
Activation
This is the description your AI agent reads to decide when to run this skill — the better it matches your request, the more reliably it fires.
Problem-solving strategies for limits colimits in category theoryKey capabilities
- →Identify limit/colimit types
- →Verify universal properties
- →Compute objects in Set
- →Formalize in Lean 4
How it works
It uses a decision tree to identify categorical structures and provides commands for symbolic computation and formal verification in Lean 4.
Inputs & outputs
When to use limits-colimits
- →Computing limits and colimits
- →Verifying universal properties
- →Formalizing category theory in Lean 4
About this skill
Limits Colimits
When to Use
Use this skill when working on limits-colimits problems in category theory.
Decision Tree
-
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.Limitsnamespace
-
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.liftgives the unique morphism
-
Colimit (Dual)
- Coproduct: colimit of discrete diagram
- Coequalizer: colimit of parallel pair
- Pushout: colimit of A <- C -> B
- Initial object: colimit of empty diagram
-
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)"
-
Preservation
- Right adjoint preserves limits
- Left adjoint preserves colimits
- Representable functors preserve limits
- Lean 4:
Adjunction.rightAdjointPreservesLimits - See:
.claude/skills/lean4-limits/SKILL.mdfor 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.
When not to use it
- →For non-categorical problems
Prerequisites
Limitations
- →Requires knowledge of category theory
- →Lean 4 verification requires formalization
How it compares
It integrates formal verification with concrete computation, unlike purely theoretical approaches.
Compared to similar skills
limits-colimits side by side with the closest alternatives in the catalog.
| Skill | Installs | Updated | Safety | Difficulty |
|---|---|---|---|---|
| limits-colimits (this skill) | 1 | 7mo | Review | Advanced |
| literature-review | 559 | 2mo | Review | Advanced |
| openalex-database | 48 | 7mo | Review | Intermediate |
| linkedin-sales-navigator-alt | 23 | 2mo | No flags | Intermediate |
Try saying
Example prompts that trigger this skill in your AI assistant.
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