[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"skill:01M6NVABGDF6CSZ76KBVC7QZAF":3,"skill-same:01M6NVABGDF6CSZ76KBVC7QZAF":56},{"skillNo":4,"name":5,"nameZh":6,"description":7,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":16,"categories":17,"keywords":18,"publishedAt":27,"linkOnly":28,"markdownContent":29,"markdownBody":30,"frontMatter":31,"refName":34,"commitSha":35,"skillDirectory":36,"forksCount":14,"repositoryHtmlUrl":37,"repositoryHomepageUrl":6,"ownerHtmlUrl":38,"ownerBlogUrl":39,"ownerDescription":40,"updatedAt":41,"install":42,"resources":55},"01M6NVABGDF6CSZ76KBVC7QZAF","limits-colimits",null,"Problem-solving strategies for limits colimits in category theory","rubicanjr\u002FFinCognis","rubicanjr","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F234912519?v=4","01M6NT156NND8J474G34SB3923","Rubi","01M6NT15B70RWXE1N3YV0BZY8A",0,"MIT","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Fcategory-theory\u002Flimits-colimits\u002FSKILL.md",[],[19,20,21,22,23,24,25,26],"limits colimits","Bash","read","problem-solving","strategies","problem","for","category theory","2026-11-10 10:12:30",false,"---\nname: limits-colimits\ndescription: \"Problem-solving strategies for limits colimits in category theory\"\nallowed-tools: [Bash, Read]\n---\n\n# Limits Colimits\n\n## When to Use\n\nUse this skill when working on limits-colimits problems in category theory.\n\n## Decision Tree\n\n\n1. **Identify Limit Type**\n   - Product: limit of discrete diagram\n   - Equalizer: limit of parallel pair f, g: A -> B\n   - Pullback: limit of A -> C \u003C- B\n   - Terminal object: limit of empty diagram\n   - Lean 4: `CategoryTheory.Limits` namespace\n\n2. **Verify Universal Property**\n   - Cone from L with projections pi_i: L -> D_i\n   - For any cone from X, unique morphism u: X -> L\n   - Triangles commute: pi_i . u = cone_i\n   - Lean 4: `IsLimit.lift` gives the unique morphism\n\n3. **Colimit (Dual)**\n   - Coproduct: colimit of discrete diagram\n   - Coequalizer: colimit of parallel pair\n   - Pushout: colimit of A \u003C- C -> B\n   - Initial object: colimit of empty diagram\n\n4. **Compute Limits Concretely**\n   - In Set: product = Cartesian product\n   - Equalizer = {x | f(x) = g(x)}\n   - Pullback = {(a,b) | f(a) = g(b)}\n   - `sympy_compute.py solve \"f(a) == g(b)\"`\n\n5. **Preservation**\n   - Right adjoint preserves limits\n   - Left adjoint preserves colimits\n   - Representable functors preserve limits\n   - Lean 4: `Adjunction.rightAdjointPreservesLimits`\n   - See: `.claude\u002Fskills\u002Flean4-limits\u002FSKILL.md` for exact syntax\n\n\n## Tool Commands\n\n### Lean4_Limit\n```bash\n# Lean 4: import CategoryTheory.Limits.Shapes.Products\n```\n\n### Lean4_Universal\n```bash\n# Lean 4: IsLimit.lift cone -- unique morphism from universal property\n```\n\n### Sympy_Pullback\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py solve \"f(a) == g(b)\"\n```\n\n### Lean4_Build\n```bash\nlake build  # Compiler-in-the-loop verification\n```\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n","# Limits Colimits\n\n## When to Use\n\nUse this skill when working on limits-colimits problems in category theory.\n\n## Decision Tree\n\n\n1. **Identify Limit Type**\n   - Product: limit of discrete diagram\n   - Equalizer: limit of parallel pair f, g: A -> B\n   - Pullback: limit of A -> C \u003C- B\n   - Terminal object: limit of empty diagram\n   - Lean 4: `CategoryTheory.Limits` namespace\n\n2. **Verify Universal Property**\n   - Cone from L with projections pi_i: L -> D_i\n   - For any cone from X, unique morphism u: X -> L\n   - Triangles commute: pi_i . u = cone_i\n   - Lean 4: `IsLimit.lift` gives the unique morphism\n\n3. **Colimit (Dual)**\n   - Coproduct: colimit of discrete diagram\n   - Coequalizer: colimit of parallel pair\n   - Pushout: colimit of A \u003C- C -> B\n   - Initial object: colimit of empty diagram\n\n4. **Compute Limits Concretely**\n   - In Set: product = Cartesian product\n   - Equalizer = {x | f(x) = g(x)}\n   - Pullback = {(a,b) | f(a) = g(b)}\n   - `sympy_compute.py solve \"f(a) == g(b)\"`\n\n5. **Preservation**\n   - Right adjoint preserves limits\n   - Left adjoint preserves colimits\n   - Representable functors preserve limits\n   - Lean 4: `Adjunction.rightAdjointPreservesLimits`\n   - See: `.claude\u002Fskills\u002Flean4-limits\u002FSKILL.md` for exact syntax\n\n\n## Tool Commands\n\n### Lean4_Limit\n```bash\n# Lean 4: import CategoryTheory.Limits.Shapes.Products\n```\n\n### Lean4_Universal\n```bash\n# Lean 4: IsLimit.lift cone -- unique morphism from universal property\n```\n\n### Sympy_Pullback\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py solve \"f(a) == g(b)\"\n```\n\n### Lean4_Build\n```bash\nlake build  # Compiler-in-the-loop verification\n```\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n",{"name":5,"description":7,"allowed-tools":32},[20,33],"Read","main","70e1c350500ee750987420bbae48837afa906b7e","skills\u002Fmath\u002Fcategory-theory\u002Flimits-colimits","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis","https:\u002F\u002Fgithub.com\u002Frubicanjr","linktr.ee\u002Ffinanskodu","Industrial Engineer & CIO @FinansKodu. Architecting AI-driven financial agents and autonomous workflows. 🏗️ Building systems, not just prompts.","2026-06-09 13:50:15",{"source":8,"skillName":5,"commands":43,"prompt":53,"zipAvailable":54},[44,47,50],{"tool":45,"command":46},"npx","npx skills add rubicanjr\u002FFinCognis --skill limits-colimits",{"tool":48,"command":49},"bunx","bunx skills add rubicanjr\u002FFinCognis --skill limits-colimits",{"tool":51,"command":52},"pnpm","pnpm dlx skills add rubicanjr\u002FFinCognis --skill limits-colimits","安装技能，技能地址：https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Fcategory-theory\u002Flimits-colimits\u002FSKILL.md",true,[],[57,78,87,106,135,154],{"skillNo":58,"name":59,"nameZh":6,"description":60,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":61,"categories":62,"keywords":68,"publishedAt":27},"01M6NWS9DAXJTADT4T5MB49X8W","workflow-router","Goal-based workflow orchestration - routes tasks to specialist agents based on user goals","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fworkflow-router\u002FSKILL.md",[63],{"categoryNo":64,"categoryName":65,"taxonomyType":66,"taxonomyTypeDesc":67},"01JBSEEDCAT000000000000023","Skill与智能体开发","SCENARIO","使用场景",[69,70,71,72,73,74,75,76,77],"workflow router","workflow","router","orchestration","goals","goal","user","workflow orchestration","tasks",{"skillNo":79,"name":80,"nameZh":6,"description":81,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":82,"categories":83,"keywords":84,"publishedAt":27},"01M6NWS3RETSC0SHP16HP66E14","wiring","Wiring Verification","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fwiring\u002FSKILL.md",[],[80,85,86],"verification","false",{"skillNo":88,"name":89,"nameZh":6,"description":90,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":91,"categories":92,"keywords":96,"publishedAt":27},"01M6NWS0XW9CVQZBDA1EGW0TWC","websocket-patterns","Connection management, room patterns, reconnection strategies, message buffering, and binary protocol design.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fwebsocket-patterns\u002Fskill.md",[93],{"categoryNo":94,"categoryName":95,"taxonomyType":66,"taxonomyTypeDesc":67},"01JBSEEDCAT000000000000015","视觉设计",[97,98,99,100,101,102,103,23,104,105],"websocket","patterns","websocket patterns","design","protocol","management","message","binary","connection",{"skillNo":107,"name":108,"nameZh":6,"description":109,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":110,"categories":111,"keywords":116,"publishedAt":27},"01M6NWRDQDX9ZNETAWRRMQWV52","visual-verdict","Screenshot comparison QA for frontend development. 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