Finn Lawler regular fibration (Rev #4)

Contents

Definition

A regular fibration is a bifibration p:EBp \colon E \to B, where EE and BB have finite products preserved by pp and by reindexing, 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

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 ff *\exists_f \dashv f^* come from pullbacks and images in CC (Elephant lemma 1.3.1) as does the Frobenius property ( op. cit. lemma 1.3.3). The terminal object of Mon(C)Mon(C) is the identity 1 1_\top on the terminal object of CC, and binary products are given by pullbacks in CC.

Revision on February 3, 2011 at 14:49:46 by Finn Lawler?. See the history of this page for a list of all contributions to it.