A regular fibration is a bifibration , where and have finite products that are preserved by and that preserve cartesian arrows, satisfying Frobenius reciprocity and the Beck--Chevalley condition for product-absolute pullback squares in .
The preservation conditions on products are equivalent to requiring that each fibre have finite products and that these be preserved by all reindexing functors .
Our regular fibrations are those of Pavlovic. A similar definition is given by Jacobs, whose only essential difference is that the fibres of a regular fibration are required to be preorders (or equivalently partial orders). Hence such ordered regular fibrations model regular logic where all proofs of the same sequent are considered equal.
A regular fibration posesses exactly the structure needed to interpret regular logic?.
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A category is regular if and only if the subobject fibration (that sends to ) is regular.
For our purposes, a regular category is one that has finite limits and pullback-stable images.
If is a regular category, then the adjunctions come from pullbacks and images in (Elephant lemma 1.3.1) as does the Frobenius property (op. cit. lemma 1.3.3). The terminal object of is the identity on , and binary products in the fibres are given by pullback. The products are preserved by reindexing functors because (the are right adjoints but also because) and have the same universal property. The projection clearly preserves these products. The Beck–Chevalley condition follows from pullback-stability of images in .
Conversely, suppose is a regular fibration. We need to show that has equalizers (to get finite limits) and pullback-stable images. But the equalizer of is . For images, let as in Elephant lemma 1.3.1. Pullback-stability follows from the Beck–Chevalley condition, which holds for all pullback squares in by Seely, Theorem sec. 8.