Overview

This Agda code accompanies our CSL 2025 paper Coslice Colimits in Homotopy Type Theory. It has been checked with Agda 2.6.3 and 2.6.4.

Organization

• HoTT-Agda/

  A stripped down version of Andrew Swan's HoTT-Agda branch, with local changes for general lemmas we proved during the development.

  It also includes wild category theory, centered on 2-coherent left adjoints. We prove that such adjoints preserve colimits (over graphs). Moreover, we prove that the Suspension functor is a 2-coherent left adjoint to Loop-space, for which we develop some new properties of homogeneous types. As a result, we get a formal proof that Suspension preserves colimits.

  See HoTT-Agda/README.md for the license of the work inside this directory.

• Colimit-code/

  Our formalization of our construction of an A-colimit.

  See Colimit-code/README.md for details and for the license of the work inside this directory.

• Pullback-stability/

  Our formalization of pullback stability (or universality) for all ordinary colimits.

  See Pullback-stability/README.md for details and for the license of the work inside this directory.

Type-checking with Docker

We have successfully tested the following Docker container on Linux but not on other operating systems.

1. Build Docker image:

   docker build . -t colimit

   The build, which uses Agda 2.6.4, type checks the whole development. The Agda code is partitioned into multiple stages across files to facilitate type-checking. The entire build may take over an hour. The type checking of all our Agda code takes about 38 minutes on our host Ubuntu with 16 GB of RAM.

2. Generate HTML files:

   mkdir -p ./html1 ./html2
   docker run --mount type=bind,source=./html1,target=/build/Colimit-code/html \
     --mount type=bind,source=./html2,target=/build/Pullback-stability/html \
     colimit

   The HTML files will be under html1/ and html2/. The entry points will be

   • html1/CosColim-main.html
   • html2/Stability.html

Acknowledgement

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-21-1-0009. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force.