Applied Category Theory Ventures

Request for Proposals                                                                    

"Theoretical mathematicians are sitting on top of a giant stockpile of intellectual gold. They have so many things that have not been manifested anywhere. You have no idea how much great stuff that these priests have in their monastery."

- Eric Weistein (Managing Director, Thiel Capital)


There are now many successful applications of category theory (CT) across many branches of mathematics and computer science (CS). A revolutionary new branch of mathematics, CT can be as transformative to information-technology (IT) as algebra has been to engineering, science, and medicine—across problems in industry, government, and society.  

At Conexus we have seen first hand, through our work on categorical data migration, that powerful mathematical results directly relevant to IT are being created faster than the ability of businesses to implement them, and our goal is to encourage the growth of a nascent, formal, and canonical “categorical operating system” upon which to build, deploy, and manage expressions of CT.  This platform is demanded not only to tackle long-standing problems in IT by newer, more powerful methods, but also to enable new technologies, such as machine learning, cloud computing, and formal verification, to be applied to IT systems in a modular, uniform, and compositional way.  

Conexus is requesting proposals for up to $1.5M USD of venture capital over up to 3 years that apply CT to business problems. These will be accepted as part of a mathematically-enabled start-up accelerator and educational institute, Conexus Ventures.  We use mixed accelerator-institute terminology because CT must be learned before it can be applied, and so Conexus maintains an educational mission as well.  A list of project ideas from the academic and industrial applied category theory community that can be developed into proposals is included at the end of this RFP.

In addition, Conexus is requesting proposals from individuals with either working knowledge of CT and a desire to apply it, or domain expertise and a desire to learn CT.  Conexus will attempt to match such individuals with appropriate applicants and/or awardees.  

Applications should be submitted via the Evalato Awards Management System no later than 11:59pm US Eastern Daylight Time (EDT) on June 30, 2019 with funding decisions planned for Aug 31, 2019.  

Clarification 1: All people are encouraged to apply as individuals, regardless of any academic affiliation.  All start-ups are welcome to apply, regardless of the genesis of their technology - in both cases, subject to the applicability of category theory in their domain.  

Clarification 2: For start-ups, the $1.5M venture capital is presumed to be equity financed but we will work with each start-up individually to define award terms.

Clarification 3: There are no style guidelines beyond this document; proposals are submitted in PDF format.

Clarification 4: Before awards are made, additional details about the proposal are likely to be required, based on reviewer feedback.


Dr. Eric Daimler served the Obama Administration as a Presidential Innovation Fellow within the Office of Science and Technology Policy ("OSTP") workflow for Robotics and Artificial Intelligence. Daimler obtained a B.S., M.S., and a PhD from the School of Computer Science ("SCS") at Carnegie Mellon University, where he later became a professor.

Dr. David Spivak is a research scientist in the MIT department of mathematics, where he uses CT to model information-bearing structures of all kinds, ranging from databases and programs to social networks and proteins. His work has been funded by NIST, NSF, AFOSR, ONR, NASA, and private companies.

Dr. Ryan Wisnesky specializes in the application of CT to query language design. He holds CS degrees from Stanford and Harvard, was a postdoc in the MIT math department, and contributed to IBM research's Clio and HIL data integration tools as well as the Coq formalization of Hedera Hashgraph.

Additional Reviewers

Dr. Brendan Fong is a postdoctoral associate in the MIT math department, following a PhD in CS at Oxford. His research explores how to use pictures to represent and reason about the systems, and how to understand the world from a relational, rather than reductionist, viewpoint. These two topics find their intersection in applied category theory.

Dr. Christina Vasilakopoulou received her PhD from University of Cambridge in 2014 in CT. Since then she has engaged in various projects in theory and applications of categories as a postdoctoral researcher, in institutions like MIT, Université Libre de Bruxelles and University of California at Riverside. Her research interests include monoidal categories, enriched category theory, categorical models of systems and networks, Hopf algebras, and more.

Applicant Qualifications

We are soliciting up to 5-page applications for up to $1.5M USD of venture capital over up to 3 years from three kinds of entities, from the U.S. and abroad:

  1. Start-up: The proposing entity should be an early stage company (or collection of individuals) with a relevant project idea, technical and business plan, and ability to execute, as described below.  In this case, venture capital will be financed by equity in the star-upt.

  1. Ongoing: The proposing entity should be an on-going concern, as a company (possibly public benefit), governmental, or academic unit, with a project currently, or quickly, in implementation that could benefit from a targeted applied category sub-project, as described below.  In this case, venture capital will be financed on a per project basis.

  1. Individual: The proposing person should have either:
  1. working knowledge of CT and a desire to apply it. 
  2. domain expertise and a desire to learn CT.

In addition, Conexus is soliciting applications with arbitrary budgets from one kind of entity:

  1. Transformational: The proposing entity is a large on-going concern, as a company (possibly public benefit), governmental, or academic unit, in an industry where the rate of progress in IT is slowing down due to data interoperability issues, with a project to solve these issues using a large-scale applied category project, possibly requiring industry-wide, governmental or academic involvement.  In this case, venture capital will be financed on a per project basis.

Category Theory

CT formalizes mathematical structure and data in terms of a labeled directed multi-graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. CT has been used to formalize traditional abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions, introduced in 1942–45 to study processes that preserve algebraic topology.  CT has contributed to CS in fundamental ways:

The Categorical Manifesto describes the conceptual clarity enabled by CT and the academic and industrial ACT community is large and growing fast.  

Categorical Query Language

A key underlying computational breakthrough enabling Conexus’s data migration technology is the open-source Categorical Query Language (“CQL”), developed jointly with MIT, the NSF (National Science Foundation), and NIST (National Institute of Standards and Technology), and being commercialized by Conexus.  CQL is an ETL tool that cleans, migrates, integrates, and validates data, using new algorithms from applied CT.  CQL’s advantages include:

Applicants are encouraged to use or extend CQL in their product roadmaps and may receive licenses to Conexus’s intellectual property to do so.

Form of the Proposal

Entities should submit a proposal based on the closest description of the entity’s type: start-up, on-going, or individual.

Proposal Format: Start-up

Start-up proposals should be less than or equal to five (5) pages long with standard margins and font, and consist of the following numbered sections:

  1. Title page

Provide a title and a ≤150-word abstract about the proposed project (an applied CT project with a maximum budget of $1.5M USD for three years). Also specify the name and email address for a point of contact at the proposing entity, as well as basic legal information about the entity (C-corp, 501(c)(3), etc).

  1. Overview

Provide a one (1) page executive summary of the project. It should describe the project, convey its importance and feasibility, and justify the teams ability to execute.

  1. Project Background, Motivation, Objectives, and Plans

Describe the project’s motivation, its technical and business objectives, and a plan to achieve the objectives.  Justify why the plan will achieve the objectives, and describe the project’s connection to applied CT.  Where applicable, describe a software development plan that uses or extends CQL to achieve the technical objectives. Provide a business development plan uniquely enabled by CT.  Provide relevant background technical, business, or other information.

  1. Deliverables, Timeline, and Budget

Propose concrete, named deliverables such as working software, proven theorems, or user engagements as well as a timeline for completing them. Describe why these deliverables and timeline implement the plan from the previous section. Describe a ≤ $1.5M budget and justify why the funds are sufficient to achieve the deliverables in the time allotted (≤ 3 years).

  1. Team

Provide a one-paragraph biographical sketch of each team member, and provide a URL to each member’s resume or CV. For each team member, explain how they would contribute to the project, and provide URLs for any relevant technical artifacts to which they have previously contributed and would like to be evaluated on.

  1. Technical or Business Requirements

Conexus recognizes that applying breakthrough mathematics to practical applications is challenging and may require research and development (R&D) that the proposing entity is not equipped to handle.  As part of this RFP, Conexus is soliciting applications from individuals without projects for the purpose of matching such R&D requirements with suitable individuals. In this section, describe any missing personnel or technical obstacles that would prevent the implementation of the proposal, including potential intellectual property issues with non-Conexus technology.

Proposal Format: On-going

On-going proposals should include the first six sections above, as well as the following:

  1. Larger Context

Conexus recognizes that research and development efforts at the proposing entity may directly or indirectly help the applied CT community, creating a positive feedback loop. Without revealing proprietary information, please describe how the proposed project fits into the proposing entity’s larger strategy and describe any possible potential synergies that may arise within the proposing entity, in connection to the applied CT community, and with any other responses to this RFP by the proposer.

Proposal Format: Transformational

Transformational proposals should follow the “On-going” format above, except there are no budget or timeline limitations.  In addition, transformational proposals must clearly describe why the problem they are solving cross-cuts their entire organization or industry, and why the proposed plan can change their entire organization or industry.  

Proposal Format: Individual

Individuals without projects should submit a 1-page document with the following sections:

  1. Name and email address
  2. Any academic degrees
  3. URL of resume or CV
  4. URLs for technical evaluation (CQL files, programs, papers, etc)
  5. A paragraph describing CT background
  6. A paragraph describing other technical / business background / domain expertise
  7. A paragraph describing the kind of project desired and how the applicant can contribute

Evaluation Criteria

Proposals will be evaluated for the team’s technical proficiency in and the plan’s technical feasibility in the following areas, as applicable (not all criteria will apply to all proposals):

  1. Category theory
  2. Mathematics
  3. Programming
  4. Other ‘areas of expertise’, such Art/Science/Engineering/Law/Medicine.
  5. CQL

Demonstrations of technical proficiency / feasibility can include, but are not limited to, CQL files, programs, theorems, papers, patents, and presentations.

Proposals will also be evaluated for the strength of the business plan:

  1. Depth of domain knowledge
  2. Relevance of domain to CT
  3. Characterization of market appropriate for CT (size, growth)
  4. Ability of team to execute
  5. Customer discovery / market data
  6. Sales & Marketing strategy
  7. Existing revenue / profit / growth / traction
  8. Synergy with other categorical components or other RFP applications

ACT Examples

In this section we briefly list some examples of ACT from the ACT community.  Solutions to problems in these areas can potentially be developed into responses to this RFP.  Responses including applications not on this list are also encouraged.

Categorical programming languages (like Haskell, Coda, CPL )

Categorical compilers (into intermediate or low-level representations, like CAML)

Categorically-specified GUIs (wiring diagrams, monoidal calculi)

Categorical statistics

Categorical dynamic systems 

Categorical knowledge representation

Categorical data science

Categorical resource theory

Categorical information theory

Categorical collaborative design

Categorical bayesian networks

Categorical Markov processes 

Categorical machine learning 

Categorical systems engineering 

Categorical mereology

Categorical reaction networks

Categorical quantum cryptography

Categorical music

Categorical internet of things

Categorical material science

Categorical nanotechnology

Categorical solution of nonlinear equations

Categorical contracts