Geodesic
The elementary theory of the 2-category of small categories
Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 196-242.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

We give an elementary description of 2-categories Cat(E) of internal categories, functors and natural transformations, where E is a category modelling Lawvere's elementary theory of the category of sets (ETCS). This extends Bourke's characterisation of 2-categories Cat(E) where 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 E give rise to generating families (resp. orthogonal factorisation systems) on Cat(E)_1, results which we believe are of independent interest.
Publié le :
Classification : 03B30, 03E30, 03G30, 18A15, 18B05, 18B25, 18B50, 18D40, 18N10
Keywords: Set theory, elementary toposes, internal categories, 2-categories, elementary theories 
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Calum Hughes; Adrian Miranda. The elementary theory of the 2-category of small categories. Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 196-242. http://geodesic.mathdoc.fr/item/TAC_2025_43_a7/