---
name: categories-functors-71b4b2bc8156
description: "Problem-solving strategies for categories functors in category theory"
allowed-tools: [Bash, Read]
---

# Categories Functors

## When to Use

Use this skill when working on categories-functors problems in category theory.

## Decision Tree


1. **Verify Category Axioms**
   - Objects and morphisms (arrows) defined?
   - Identity morphism for each object: id_A: A -> A
   - Composition associative: (f . g) . h = f . (g . h)
   - Write Lean 4: `theorem assoc : (f ≫ g) ≫ h = f ≫ (g ≫ h) := Category.assoc`

2. **Check Functor Properties**
   - F: C -> D maps objects to objects, arrows to arrows
   - Preserves identity: F(id_A) = id_{F(A)}
   - Preserves composition: F(g . f) = F(g) . F(f)
   - Write Lean 4: `theorem comp : F.map (g ≫ f) = F.map g ≫ F.map f := F.map_comp`

3. **Functor Types**
   - Covariant: preserves arrow direction
   - Contravariant: reverses arrow direction
   - Faithful/Full: injective/surjective on Hom-sets
   - Equivalence: full, faithful, essentially surjective

4. **Common Functors**
   - Forgetful functor: forgets structure (e.g., Grp -> Set)
   - Free functor: left adjoint to forgetful
   - Hom functor: Hom(A, -) or Hom(-, B)
   - Power set functor: Set -> Set via X |-> P(X)

5. **Verify with Lean 4**
   - Compiler-in-the-loop: write proof, `lake build` checks
   - Mathlib has full category theory library
   - See: `.claude/skills/lean4-functors/SKILL.md` for exact syntax


## Tool Commands

### Lean4_Category
```bash
# Lean 4 with Mathlib: import CategoryTheory.Category.Basic
```

### Lean4_Functor
```bash
# Lean 4: theorem map_comp (F : C ⥤ D) : F.map (g ≫ f) = F.map g ≫ F.map f := F.map_comp
```

### Lean4_Build
```bash
lake build  # Compiler-in-the-loop verification
```

## Cognitive Tools Reference

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


---
Source attribution: vibeeval
Original project: https://github.com/vibeeval/vibecosystem/blob/3b763b1fb288f57bfa3cce76ef18184b96461a78/skills/math/category-theory/categories-functors/SKILL.md
License: MIT
License information: https://opensource.org/license/mit
Original upstream instructions; downloaded from Mega AI Hub.

## Upstream license notice

MIT License

Copyright (c) 2026 vibeeval

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# Mega AI Hub resource metadata

Creator / publisher: vibeeval
Original project: https://github.com/vibeeval/vibecosystem/blob/3b763b1fb288f57bfa3cce76ef18184b96461a78/skills/math/category-theory/categories-functors/SKILL.md
Attribution basis: Source repository record

Fields of work: Data & Analytics, Development & Engineering, Education & Research
Roles: Data Scientist, ML / AI Engineer, Researcher
Specialties: API
Tools: Lean4
Country applicability: Country not specified
Classification basis: Source catalog labels
Instruction SHA-256: 15dc40b8f51338da2e4936faaa4c902c8c9576211ba0abed3e872ad7a8a8d659