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arXiv · 2403.03647

The elementary theory of the 2-category of small categories

Calum Hughes·Adrian Miranda
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Abstract

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.

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Calum Hughes, Adrian Miranda. 2024-03-06. The elementary theory of the 2-category of small categories. https://arxiv.org/abs/2403.03647

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