Finn Lawler regular fibration (Rev #11, changes)

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Definition

A pre-regular fibration is a bifibration p:EBp \colon E \to B, where EE and BB have finite products that are preserved by pp and that preserve cartesian arrows, satisfying Frobenius reciprocity and the Beck--Chevalley condition for product-absolute pullback squares of types C, D and E in BB. A regular fibration is a pre-regular fibration that also satisfies the Beck–Chevalley condition for type-B squares.

The preservation conditions on products are equivalent to requiring that each fibre E AE_A have finite products and that these be preserved by all reindexing functors f *f^*.

The type-A Beck–Chevalley condition

Here and in my thesis I originally required regular fibrations to satisfy the type-A BC condition, but I now believe that it holds automatically as long as E does.

By Shulman’s construction, a pre-regular fibrationEEShulman’s construction , over a pre-regular fibration B E B E gives over rise to a cartesian equipmentBMatr(E) B \to Matr(E) , and gives hence rise to a cartesian bicategory equipmentBMatr(E) B \to Matr(E) . , Type-E and squares hence in a the cartesian bicategory are exact because they satisfy the BC condition in the fibration.Matr(E)Matr(E) . then Type-E becomes squares compact in closed the bicategory, bicategory which are is exact to because say they that satisfy there the is BC an condition equivalence in the fibration. hom Matr( A E×B,C)hom(A,C×B) hom(A Matr(E) \times B, C) \simeq hom(A, C \times B) given then by becomes composing compact with closed bicategory, which is to say that there is an equivalencehom(A×d Be ,C)hom(A,C×B) A hom(A \times d_\bullet B, e^\bullet C) \simeq hom(A, C \times B) (left given to by right) composing and with C A×d d e e C A \times d^\bullet d_\bullet e_\bullet e^\bullet (right (left to left), right) where and with d C×d e d C \times d^\bullet e_\bullet is (right the to diagonal left), at where B d B d and is the diagonal at e B e B the and map from B e B e to the terminal map object. from We will write this asR :AC×B R^\wedge \colon A \to C \times B for to the terminal object. We will write this asRR :A×BC×B R R^\wedge \colon A \times B \to C \times B and forS R:A×BC S^\vee R \colon A \times B \to C for andSS :AC×BC S S^\vee \colon A \to C \times B \to C . Similarly for there is an equivalence hom S ( :A ×B,C)hom(B,A× C B) hom(A S \times \colon B, C) \simeq hom(B, A \to C \times C) B , . which Similarly we there will is write an as equivalence homT(A×B,C)hom(B,A×C) {}^\wedge hom(A T \times B, C) \simeq hom(B, A \times C) , and which we will write as U T {}^\vee {}^\wedge U T . If andR :UAB R {}^\vee \colon U A \to B . then IfR R = : A (R B) (R ) R^\circ R = \colon {}^\vee A (R^\wedge) \to \cong B {}^\wedge (R^\vee) . The then type-E BC condition implies thatf R =f (R ) (R ) f_\bullet R^\circ \dashv = f_\bullet^\circ {}^\vee (R^\wedge) \cong {}^\wedge (R^\vee) , . and The therefore type-E BC condition implies thatf f f f_\bullet \dashv f_\bullet^\circ \cong f^\bullet , and also therefore that d f d f d_\bullet^\vee f_\bullet^\circ \cong d^\bullet f^\bullet , and also that(d d ) d d (d^\bullet)^\wedge d_\bullet^\vee \cong d_\bullet d^\bullet . and(d ) d (d^\bullet)^\wedge \cong d_\bullet. (See Carboni–Walters and Walters–Wood for all of this.)

The claim now is that modulo these isomorphisms, the morphisms required to be invertible by exactness of or BC for type-A squares are equal to β 1=(d f (f×f) d ) \beta_1 = (d_\bullet f_\bullet \cong (f \times f)_\bullet d_\bullet)^\vee and β 2= β 1\beta_2 = {}^\wedge \beta_1. Being the images under equivalences of invertible morphisms, then, they must themselves be invertible.

The following pictures show the essential parts of the proof using string diagrams, which can be read top-to-bottom as morphisms in a cartesian bicategory or bottom-to-top as objects in a monoidal bifibration as in this paper.

Logic

A regular fibration posesses exactly the structure needed to interpret regular logic, i.e. the {\exists}{\wedge}{\top}-fragment of first-order logic.

Characterization

Proposition

A category CC is regular if and only if the subobject fibration Sub(C)CSub(C) \to C (that sends SXS \hookrightarrow X to XX) is regular.

Proof

For our purposes, a regular category is one that has finite limits and pullback-stable images.

If CC is a regular category, then 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 Sub(A)=Sub(C) ASub(A) = Sub(C)_A is the identity 1 A1_A on AA, and binary products in the fibres Sub(A)Sub(A) are given by pullback. The products are preserved by reindexing functors f *f^* because (the f *f^* are right adjoints but also because) f *(SS)f^*(S \wedge S') and f *Sf *Sf^*S \wedge f^*S' have the same universal property. The projection Sub(C)CSub(C) \to C clearly preserves these products. The Beck–Chevalley condition follows from pullback-stability of images in CC.

Conversely, suppose Sub(C)CSub(C) \to C is a regular fibration. We need to show that CC has equalizers (to get finite limits) and pullback-stable images. But the equalizer of f,g:ABf, g \colon A \rightrightarrows B is (f,g) *Δ(f,g)^*\Delta. For images, let imf= f1im f = \exists_f 1 as in Elephant lemma 1.3.1. Pullback-stability follows from the Beck–Chevalley condition, which holds for all pullback squares in CC by Seely, Theorem sec. 8.

Revision on December 9, 2015 at 17:37:59 by Finn Lawler?. See the history of this page for a list of all contributions to it.