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Definition

A pre-regular fibration is a bifibration $p \colon E \to B$, where $E$ and $B$ have finite products that are preserved by $p$ 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 $B$. 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 $E_A$ have finite products and that these be preserved by all reindexing functors $f^*$.

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 fibration$E$Shulman’s construction , over a pre-regular fibration$\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>}$ B E gives over rise to a cartesian equipment$B\to \mathrm{Matr}(E)$ B \to Matr(E) , and gives hence rise to a cartesian bicategory equipment$B\to \mathrm{Matr}(E)$ 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.$Matr(E)$ . 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.$\mathrm{<span><del\; class="diffmod">\; hom</del><ins\; class="diffmod">\; Matr</ins></span>}(\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; E</ins></span>}\times B,C)\simeq \mathrm{hom}(A,C\times B)$ 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 equivalence$\mathrm{hom}(A\times d_{•}B{e}^{•},C)\simeq \mathrm{hom}(A,C\times B)$ A hom(A \times d_\bullet B, e^\bullet C) \simeq hom(A, C \times B) (left given to by right) composing and with$\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; A</ins></span>}\times d^{•}{d}_{•}{e}_{•}{e}^{•}$ C A \times d^\bullet d_\bullet e_\bullet e^\bullet (right (left to left), right) where and with$\mathrm{<span><del\; class="diffmod">\; d</del><ins\; class="diffmod">\; C</ins></span>}\times d^{•}{e}_{•}$ d C \times d^\bullet e_\bullet is (right the to diagonal left), at where$\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; d</ins></span>}$ B d and is the diagonal at$\mathrm{<span><del\; class="diffmod">\; e</del><ins\; class="diffmod">\; B</ins></span>}$ e B the and map from$\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; e</ins></span>}$ B e to the terminal map object. from We will write this as$R^{\wedge }:A\to C\times B$ R^\wedge \colon A \to C \times B for to the terminal object. We will write this as$R{R}^{\wedge }:A\times B\to C\times B$ R R^\wedge \colon A \times B \to C \times B and for$S^{\vee }R:A\times B\to C$ S^\vee R \colon A \times B \to C for and$S{S}^{\vee }:A\to C\times B\to C$ S S^\vee \colon A \to C \times B \to C . Similarly for there is an equivalence$\mathrm{<span><del\; class="diffmod">\; hom</del><ins\; class="diffmod">\; S</ins></span>}<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; :</ins></span>A<span><del\; class="diffmod">\; \times </del><ins\; class="diffmod">\; \to </ins></span>B,C)\simeq \mathrm{hom}(B,A\times \mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; B</ins></span>})$ 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${}^{\wedge }\mathrm{hom}T(A\times B,C)\simeq \mathrm{hom}(B,A\times C)$ {}^\wedge hom(A T \times B, C) \simeq hom(B, A \times C) , and which we will write as${}^{<span><del\; class="diffmod">\; \vee </del><ins\; class="diffmod">\; \wedge </ins></span>}\mathrm{<span><del\; class="diffmod">\; U</del><ins\; class="diffmod">\; T</ins></span>}$ {}^\vee {}^\wedge U T . If and$R{}^{\vee }:UA\to B$ R {}^\vee \colon U A \to B . then If$R^{\circ }R<span><del\; class="diffmod">\; =</del><ins\; class="diffmod">\; :</ins></span>{}^{\vee }A<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; \to </ins></span>R^{\wedge }B)\cong {}^{\wedge }(R^{\vee })$ R^\circ R = \colon {}^\vee A (R^\wedge) \to \cong B {}^\wedge (R^\vee) . The then type-E BC condition implies that$f_{•}{R}^{\circ }<span><del\; class="diffmod">\; ⊣</del><ins\; class="diffmod">\; =</ins></span>f_{•}^{\circ }{}^{\vee }(R^{\wedge })\cong {}^{\wedge }(R^{\vee })$ f_\bullet R^\circ \dashv = f_\bullet^\circ {}^\vee (R^\wedge) \cong {}^\wedge (R^\vee) , . and The therefore type-E BC condition implies that$f_{•}⊣f_{•}^{\circ }\cong f^{•}$ f_\bullet \dashv f_\bullet^\circ \cong f^\bullet , and also therefore that${\mathrm{<span><del\; class="diffmod">\; d</del><ins\; class="diffmod">\; f</ins></span>}}_{•}^{<span><del\; class="diffmod">\; \vee </del><ins\; class="diffmod">\; \circ </ins></span>}\cong {\mathrm{<span><del\; class="diffmod">\; d</del><ins\; class="diffmod">\; f</ins></span>}}^{•}$ d_\bullet^\vee f_\bullet^\circ \cong d^\bullet f^\bullet , and also that$(d_{•}^{\vee }{d}^{•}{)}^{\wedge }\cong d_{•}{d}^{•}$ (d^\bullet)^\wedge d_\bullet^\vee \cong d_\bullet d^\bullet . and$(d^\bullet)^\wedge \cong d_\bullet$. (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 $\beta_1 = (d_\bullet f_\bullet \cong (f \times f)_\bullet d_\bullet)^\vee$ and $\beta_2 = {}^\wedge \beta_1$. 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.

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Logic

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

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Characterization