A regular fibration is a bifibration , where and have finite products preserved by and by reindexing, satisfying Frobenius reciprocity and the Beck--Chevalley condition for pullback squares in .
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 projection sending to is a regular fibration, where is the full subcategory of on the monomorphisms.
For our purposes, a regular category is one that has finite limits? and pullback?-stable images?.
The ‘only if’ part of the proposition is straightforward: 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 the terminal object of , and binary products are given by pullbacks in .
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