catcolab-schemas

# CatColab Schemas: Database Schema Modeling **Trit**: +1 (PLUS - generator) **Color**: Green (#32CD32) ## Overview Schemas in CatColab upgrade ologs by explicitly distinguishing: - **Entities**: Tables with identity (foreign key targets) - **Attributes**: Columns/properties (data values) - **Mappings**: Foreign key relationships This is the foundation for **ACSets** (Attributed C-Sets), the core data structure of AlgebraicJulia. ## Mathematical Foundation A schema is a **profunctor** or displayed category: ``` ┌─────────────────────────────────────────────────────┐ │ SCHEMA │ ├─────────────────────────────────────────────────────┤ │ Entities (Ob): │ │ Person, Company, Project │ │ │ │ AttrTypes (Data): │ │ String, Int, Date, Bool │ │ │ │ Mappings (Hom): │ │ works_at: Person → Company │ │ leads: Person → Project │ │ │ │ Attributes (Attr): │ │ name: Person → String │ │ age: Person → Int │ │ founded: Company → Date │ └─────────────────────────────────────────────────────┘ ``` ## Double Theory ```rust // Schema double theory in catlog pub fn th_schema() -> DiscreteDblTheory { let mut cat = FpCategory::new(); // Object types cat.add_ob_generator(name("Entity")); cat.add_ob_generator(name("AttrType")); // Morphism types cat.add_mor_generator(name("Mapping"), name("Entity"), name("Entity")); cat.add_mor_generator(name("Attr"), name("Entity"), name("AttrType")); cat.into() } ``` ## CatColab Implementation ### Entity Declaration ```typescript { "type": "ObDecl", "name": "Person", "theory_type": "Entity", "description": "people in the system" } ``` ### Attribute Type Declaration ```typescript { "type": "ObDecl", "name": "String", "theory_type": "AttrType", "description": "text values" } ``` ### Mapping (Foreign Key) ```typescript { "type": "MorDecl", "name": "employer", "dom": "Person", "cod": "Company", "theory_type": "Mapping", "description": "the company this person works for" } ``` ### Attribute (Column) ```typescript { "type": "MorDecl", "name": "salary", "dom": "Person", "cod": "Int", "theory_type": "Attr", "description": "annual salary in dollars" } ``` ## ACSet Connection A CatColab schema defines the type; an **ACSet** is an instance: ```julia # Schema defines structure @present SchPerson(FreeSchema) begin Person::Ob Company::Ob employer::Hom(Person, Company) Name::AttrType name::Attr(Person, Name) end # ACSet populates data people = @acset SchPerson begin Person = 3 Company = 2 employer = [1, 1, 2] name = ["Alice", "Bob", "Charlie"] end ``` ## Practical Examples ### Example 1: E-Commerce Schema ``` Entities: Customer, Order, Product, Category AttrTypes: String, Int, Float, DateTime Mappings: placed_by: Order → Customer contains: Order → Product (many-to-many via junction) in_category: Product → Category Attributes: email: Customer → String total: Order → Float price: Product → Float name: Category → String ``` ### Example 2: Social Network ``` Entities: User, Post, Comment, Group AttrTypes: String, DateTime, Int Mappings: author: Post → User on_post: Comment → Post member_of: User → Group Attributes: username: User → String content: Post → String timestamp: Post → DateTime likes: Post → Int ``` ## Schema Composition Schemas compose via **pullback** and **pushout**: ``` Schema A Schema B \ / \ pullback / \ / ▼ ▼ Schema A ×_C B ``` ## GF(3) Triads ``` catcolab-ologs (-1) ⊗ topos-catcolab (0) ⊗ catcolab-schemas (+1) = 0 ✓ database-design (-1) ⊗ acsets-relational-thinking (0) ⊗ catcolab-schemas (+1) = 0 ✓ ``` ## Commands ```bash # Create schema just catcolab-new schema "my-database" # Generate Julia ACSet code just catcolab-export my-database --format=julia # Create instance (diagram) just catcolab-instance my-database "sample-data" # Migrate schema just catcolab-migrate old-schema new-schema ``` ## References - Patterson et al. "Categorical data structures for technical computing" (2022) - [AlgebraicJulia ACSets](https://algebraicjulia.github.io/ACSets.jl/) - [CatColab Schema Help](https://catcolab.org/help/logics/schema) --- **Skill Name**: catcolab-schemas **Type**: Database Schema Design **Trit**: +1 (PLUS) **GF(3)**: Conserved via triadic composition