We present a doctrinal approach to category theory, obtained by abstracting from the indexed inclusion (via discrete fibrations and opfibrations) of left and of right actions of X in Cat in categories over X. Namely, a ``weak temporal doctrine'' consists essentially of two indexed functors with the same codomain such that the induced functors have both left and right adjoints satisfying some exactness conditions, in the spirit of categorical logic.
The derived logical rules include some adjunction-like laws involving the truth-values-enriched hom and tensor functors, which condense several basic categorical properties and display a nice symmetry. The symmetry becomes more apparent in the slightly stronger context of ``temporal doctrines'', which we initially treat and which include as an instance the inclusion of lower and upper sets in the parts of a poset, as well as the inclusion of left and right actions of a graph in the graphs over it.
Keywords: Temporal doctrine; internally enriched hom, tensor and (co)limits; Frobenius law; adjunction-like laws; quantification formulas
2000 MSC: 18A05, 18A30, 18A40, 18D99
Theory and Applications of Categories,
Vol. 24, 2010,
No. 15, pp 394-417.