Finn Lawler regular fibration (Rev #3)

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Contents

Definition

A regular fibration is a bifibration p:EBp \colon E \to B, where EE and BB have finite products preserved by pp, satisfying Frobenius reciprocity and the Beck--Chevalley condition for pullback squares in BB.

Logic

A regular fibration posesses exactly the structure needed to interpret regular logic?.

Characterization

Proposition

A category CC is regular if and only if the projection cod:Mon(C)Ccod \colon Mon(C) \to C sending SXS \hookrightarrow X to XX is a regular fibration, where Mon(C)Mon(C) is the full subcategory of C C^{\to} on the monomorphisms.

Proof

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