A logic for categories

TL;DR

This paper introduces a doctrinal framework for category theory using indexed functors with adjoints, revealing symmetrical logical rules and properties in categorical structures.

Contribution

It develops the concept of weak and temporal doctrines, providing a new logical perspective on categorical actions and inclusions with adjoint functors.

Findings

Derived logical rules include adjunction-like laws with symmetry.

Includes examples like lower and upper sets in posets.

Shows a unified approach to categorical actions and inclusions.

Abstract

We present a doctrinal approach to category theory, obtained by abstracting from the indexed inclusions (via discrete fibrations and opfibrations) of the left and of the 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 display a nice symmetry and condense several basic categorical properties. 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.