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**Published in:** LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Quillen model category theory forms the cornerstone of modern homotopy theory, and thus the semantics of (and justification for the name of) homotopy type theory / univalent foundations (HoTT/UF). One of the main tools of Quillen model category theory is the small object argument. Indeed, the particular model categories that can interpret HoTT/UF are usually constructed using the small object argument.
In this article, we formalize the algebraic small object argument, a modern categorical version of the small object argument originally due to Garner, in the Coq UniMath library. This constitutes a first step in building up the tools required to formalize - in a system based on HoTT/UF - the semantics of HoTT/UF in particular model categories: for instance, Voevodsky’s original interpretation into simplicial sets.
More specifically, in this work, we rephrase and formalize Garner’s original formulation of the algebraic small object argument. We fill in details of Garner’s construction and redefine parts of the construction to be more direct and fit for formalization. We rephrase the theory in more modern language, using constructions like displayed categories and a modern, less strict notion of monoidal categories. We point out the interaction between the theory and the foundations, and motivate the use of the algebraic small object argument in lieu of Quillen’s original small object argument from a constructivist standpoint.

Dennis Hilhorst and Paige Randall North. Formalizing the Algebraic Small Object Argument in UniMath. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hilhorst_et_al:LIPIcs.ITP.2024.20, author = {Hilhorst, Dennis and North, Paige Randall}, title = {{Formalizing the Algebraic Small Object Argument in UniMath}}, booktitle = {15th International Conference on Interactive Theorem Proving (ITP 2024)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-337-9}, ISSN = {1868-8969}, year = {2024}, volume = {309}, editor = {Bertot, Yves and Kutsia, Temur and Norrish, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.20}, URN = {urn:nbn:de:0030-drops-207486}, doi = {10.4230/LIPIcs.ITP.2024.20}, annote = {Keywords: formalization of mathematics, univalent foundations, model category theory, algebraic small object argument, coq, unimath} }

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**Published in:** LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

We combine the theory of inductive data types with the theory of universal measurings. By doing so, we find that many categories of algebras of endofunctors are actually enriched in the corresponding category of coalgebras of the same endofunctor. The enrichment captures all possible partial algebra homomorphisms, defined by measuring coalgebras. Thus this enriched category carries more information than the usual category of algebras which captures only total algebra homomorphisms. We specify new algebras besides the initial one using a generalization of the notion of initial algebra.

Paige Randall North and Maximilien Péroux. Coinductive Control of Inductive Data Types. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{north_et_al:LIPIcs.CALCO.2023.15, author = {North, Paige Randall and P\'{e}roux, Maximilien}, title = {{Coinductive Control of Inductive Data Types}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {15:1--15:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.15}, URN = {urn:nbn:de:0030-drops-188129}, doi = {10.4230/LIPIcs.CALCO.2023.15}, annote = {Keywords: Inductive types, enriched category theory, algebraic data types, algebra, coalgebra} }

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