The elementary theory of the 2-category of small categories
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912291523395584 |
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| author | Hughes, Calum Miranda, Adrian |
| author_facet | Hughes, Calum Miranda, Adrian |
| contents | We give an elementary description of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ of internal categories, functors and natural transformations, where $\mathcal{E}$ is a category modelling Lawvere's elementary theory of the category of sets (ETCS). This extends Bourke's characterisation of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ where $\mathcal{E}$ has pullbacks to take account for the extra properties in ETCS, and Lawvere's characterisation of the (one dimensional) category of small categories to take account of the two-dimensional structure. Important two-dimensional concepts which we introduce include $2$-well-pointedness, full-subobject classifiers, and the categorified axiom of choice. Along the way, we show how generating families (resp. orthogonal factorisation systems) on $\mathcal{E}$ give rise to generating families (resp. orthogonal factorisation systems) on $\mathbf{Cat}\left(\mathcal{E}\right)_{1}$, results which we believe are of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03647 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The elementary theory of the 2-category of small categories Hughes, Calum Miranda, Adrian Category Theory Logic 03B30, 03E30, 03G30, 18A15, 18B05, 18B25, 18B50, 18D40, 18N10 We give an elementary description of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ of internal categories, functors and natural transformations, where $\mathcal{E}$ is a category modelling Lawvere's elementary theory of the category of sets (ETCS). This extends Bourke's characterisation of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ where $\mathcal{E}$ has pullbacks to take account for the extra properties in ETCS, and Lawvere's characterisation of the (one dimensional) category of small categories to take account of the two-dimensional structure. Important two-dimensional concepts which we introduce include $2$-well-pointedness, full-subobject classifiers, and the categorified axiom of choice. Along the way, we show how generating families (resp. orthogonal factorisation systems) on $\mathcal{E}$ give rise to generating families (resp. orthogonal factorisation systems) on $\mathbf{Cat}\left(\mathcal{E}\right)_{1}$, results which we believe are of independent interest. |
| title | The elementary theory of the 2-category of small categories |
| topic | Category Theory Logic 03B30, 03E30, 03G30, 18A15, 18B05, 18B25, 18B50, 18D40, 18N10 |
| url | https://arxiv.org/abs/2403.03647 |