Coslice Colimits in Homotopy Type Theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918404660658176 |
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| author | Hart, Perry Hou, Kuen-Bang |
| author_facet | Hart, Perry Hou, Kuen-Bang |
| contents | We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between (graph-indexed) colimits in a type universe and colimits in coslices of the universe, called coslice colimits. To derive this characterization, we give a construction of coslice colimits that is tailored to reveal the connection. We use the construction to prove that the forgetful functor from a coslice creates colimits over trees. We also use it to study how coslice colimits interact with orthogonal factorization systems and with cohomology theories. As a result of their interaction with orthogonal factorization systems, all colimits of pointed types preserve $n$-connectedness, which implies that higher groups, in the sense of Buchholtz, van Doorn, and Rijke, are closed under colimits. We have formalized major portions of this work (see https://github.com/PHart3/colimits-agda for the Agda code), including our main construction of the coslice colimit functor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_15103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coslice Colimits in Homotopy Type Theory Hart, Perry Hou, Kuen-Bang Logic in Computer Science Category Theory Logic We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between (graph-indexed) colimits in a type universe and colimits in coslices of the universe, called coslice colimits. To derive this characterization, we give a construction of coslice colimits that is tailored to reveal the connection. We use the construction to prove that the forgetful functor from a coslice creates colimits over trees. We also use it to study how coslice colimits interact with orthogonal factorization systems and with cohomology theories. As a result of their interaction with orthogonal factorization systems, all colimits of pointed types preserve $n$-connectedness, which implies that higher groups, in the sense of Buchholtz, van Doorn, and Rijke, are closed under colimits. We have formalized major portions of this work (see https://github.com/PHart3/colimits-agda for the Agda code), including our main construction of the coslice colimit functor. |
| title | Coslice Colimits in Homotopy Type Theory |
| topic | Logic in Computer Science Category Theory Logic |
| url | https://arxiv.org/abs/2411.15103 |