Categorical Databases

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The suggested introductory course of study includes an introductory paper about CQL for computer scientists, introductory slides and video about CQL from a functional programming perspective, introductory slides and video about CQL from a mathematical perspective, introductory slides about category theory from a knowledge management perspective, a paper from NIST arguing that category theory is critical for IT interoperability, and an introductory textbook on category theory, as well as a data science case study.

Papers

• Ologs: A Categorical Framework for Knowledge Representation (2011) [Pre-print]

  Describes how to use categories as database/ontology schemas.

• Functorial Data Migration (2013) [Pre-print]

Describes how to migrate data between categorical databases.

• Database Queries and Constraints via Lifting Problems (2013) [Pre-print]

Describes how certain common database queries and constraints can be encoded as topological lifts.

• Definition of CQL as a context-free grammar with equations (2015)

Describes a canonical syntax associated with CQL, as well as an axiomatic semantics.

• Relational Foundations for Functorial Data Migration (2015) [Pre-print] [slides]

Describes how to implement a fragment of CQL using SELECT/FROM/WHERE/UNION and a fresh-ID generator, and vice versa.

• QINL: Query-Integrated Languages (2015)

Describes the relationship between CQL and comprehension/monad syntax.

• Algebraic Model Management: A Survey (2016) [Pre-print]

Describes CQL entirely in terms of multi-sorted equational logic, and contrasts it with existing tools.

• Algebraic Databases (2017) [Pre-print]

Describes CQL in detail, and in particular, user-defined functions.

• Algebraic Data Integration (2017) [Pre-print] [slides] [Aggregation supplement] [video]

Describes CQL in detail, and in particular, how to implement it using automated theorem proving techniques and how to use it to integrate data.

• Informal Data Transformation Considered Harmful (2019)

Describes how CQL can preserve the data quality required to power machine learning algorithms through various data management tasks.

• Fast Left Kan Extensions Using the Chase (2022)

Describes CQL's sigma operation in detail, and in particular, how to implement it using a chase engine.

• Presenting Profunctors (2024)

Motivated by problems in categorical database theory, we introduce and compare two notions of presentation for profunctors, uncurried and curried.

Case Studies

• Using Category Theory to Facilitate Multiple Manufacturing Service Database Integration (2017) [Pre-print] [Companion Report]

Describes how CQL can be used for ontology-driven semantic search, and applied to commercial supply chains. Joint work with NIST, it extends the RDF-based approach previously pursued by NIST.

• Categorical Data Integration for Computational Science (2019) [Pre-print] [Code] [Slides]

Describes how CQL can be used to integrate scientific data sets, such as those in quantum chemistry. Joint work with Stanford.

• Compositional Models for Power Systems (2019) [Video] [Slides]

Describes how CQL can be used to integrate mathematical models of power grids in a modular way. Independent work by NIST.

• Financial Reporting Data Warehousing with CQL (2019)

Describes how CQL can be used to construct data warehouses suitable for client reporting in financial asset management.

• Algebraic Property Graphs (2019) [Coq code] [CQL code] [Java code]

Describes how CQL can be used to integrate enterprise knowledge graphs. Joint work with Uber.

• Relational to RDF Data Migration by Query Co-Evaluation (Draft, 2021)

Describes how CQL can be used to migrate relational data to RDF form, using the FIBO financial RDF ontology as an example.

• Consensus-Free Spreadsheet Integration (Draft, 2022)

Describes how CQL can be used to integrate spreadsheets. Join work with Chevron.

Presentations

• Boston Haskell (2011) [videos 1 2 3 4 5]

Gives an introduction to the math behind CQL.

• Oracle (2014)

Gives an introduction to the math behind CQL, tailored to database-centric audiences.

• Boston Haskell (2014) [video]

Gives an introduction to the math behind CQL, tailored to functional-programming audiences.

• Lambda Conf (2017) [video]

Gives an introduction to CQL for computer scientists.

• The Broad Institute (2017) [video]

Gives an introduction to the math behind CQL, and lessons learned from applying it in practice.

• Dataversity Architecture Summit (2017)

Gives an introduction to CQL tailored to data architects.

• Kensho (2019) [video]

Gives an introduction CQL and the math behind it, and describes how it enables universal semantic IT inter-operability.

Related

• A Categorical Manifesto (1991)

Describes why category theory matters: how it is useful in computer science.

• A Model Theory for Generic Schema Management (2003)

Describes how category theory can be used to study schema mappings in the sense of traditional database theory.

• Formal Modelling and Application of Graph Transformations in the Resource Description Framework (2009)

Describes various categories of RDF graphs in functorial-semantics style.

• Allegories for Database Modeling (2013)

Describes a relational variation of the math behind CQL based on allegories rather than categories.

• Sketches as a Framework for Knowledge Management (2014)

Describes how CQL and related Sketch-based formalisms can be used to integrate knowledge representations.

• Entity-Attribute Sketches (2015)

Describes EASIK, a category-theoretic predecessor to CQL that now ships within, and interoperates with, CQL.

• Knowledge Representation in Bicategories of Relations (2017)

Describes the math behind the allegorical approach to database modeling.

• Category Theory Framework for Variability Models with Non-functional Requirements (2021)

Describes an application of CQL to software engineering. [video]

• Representing Knowledge and Querying Data using Double-Functorial Semantics (2024)

Describes how the abstract structure of a 'double category of relations' is a flexible and expressive language in which to represent knowledge.