Page 111 - Textos de Matemática Vol. 45

9.3. Fibrations 103
Proposition 9.2.1. Suppose the regular epimorphisms are stable under pullbacks in E. Consider the following commutative diagram where both the whole rect- angle and the left hand side square are pullbacks:
• // // • // • // // //
•••
Then so is the right hand side one, provided that the lower left hand side hori- zontal arrow is a regular epimorphism.
We have also the following partial converse to Corollary 9.1.2, which is known as the Barr-Kock Theorem, see [1]:
Theorem 9.2.2. Suppose the regular epimorphisms are stable under pullbacks in E. Consider the following commutative diagram, where any of the left hand side squares is a pullback:
p1 R(x)
p0
// f R[f]oo s0 // X // // Y
xy

// Y ′.
regular epimorphism. Moreover, when x is a monomorphism, so is y. 9.3 Fibrations
We recall here from [2] the notions of cartesian morphisms and fibrations.
Definition 9.3.1. Let F : D → E be a functor. Given an object E ∈ E, the fiber of F at E is the subcategory FE of D whose objects are the objects D ∈ D such that F(D) = E and whose arrows are the arrows f: D → D′ in D such that F(f) = 1E.
Definition 9.3.2. Let F: D → E be a functor and α: J → E a morphism in E. A morphism f : Y → X in D is cartesian over α if F (f ) = α and, for any g: Z → X in D such that F(g) factors as F(g) = αβ for some β in E, there existsauniquemorphismh:Z→Y suchthatF(h)=βandg=fh.
Definition 9.3.3. A functor F : D → E is a fibration when, for every morphism α: J → E and every object X in the fiber FE, there exists in D a cartesian morphism over α.
′oop′0
R [ f ] s ′0 // X′
//
Then the right hand side one is also a pullback, provided that the map f is a
p′ 1
f′