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DOI: 10.4230/LIPIcs.ITP.2025.20

Formalizing Colimits in 𝒞at

Authors: Mario Carneiro and Emily Riehl

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract

Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite ∞-category theory has not been formalized. To support future work on formalizing ∞-category theory in Lean’s mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that 𝒞at has colimits. To our knowledge this is the first formalized proof that the nerve functor is a fully faithful right adjoint and that the category of categories is cocomplete.

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Mario Carneiro and Emily Riehl. Formalizing Colimits in 𝒞at. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carneiro_et_al:LIPIcs.ITP.2025.20,
  author =	{Carneiro, Mario and Riehl, Emily},
  title =	{{Formalizing Colimits in 𝒞at}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.20},
  URN =		{urn:nbn:de:0030-drops-246186},
  doi =		{10.4230/LIPIcs.ITP.2025.20},
  annote =	{Keywords: category theory, infinity-category theory, nerve, simplicial set, colimit}
}

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Formalizing Colimits in 𝒞at

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