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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.17769 |
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| _version_ | 1866915600853368832 |
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| author | Hughes, Calum Miranda, Adrian |
| author_facet | Hughes, Calum Miranda, Adrian |
| contents | We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the $2$-category $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations has finite $2$-colimits. In addition, $\mathbf{Cat}(\mathcal{E})$ is extensive, has pullbacks and codescent coequalisers are stable under pullback along discrete Conduché fibrations. Moreover, we give converse results to this. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17769 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Colimits of internal categories Hughes, Calum Miranda, Adrian Category Theory 18D40 (Primary), 18N10 (Secondary) We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the $2$-category $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations has finite $2$-colimits. In addition, $\mathbf{Cat}(\mathcal{E})$ is extensive, has pullbacks and codescent coequalisers are stable under pullback along discrete Conduché fibrations. Moreover, we give converse results to this. |
| title | Colimits of internal categories |
| topic | Category Theory 18D40 (Primary), 18N10 (Secondary) |
| url | https://arxiv.org/abs/2501.17769 |