Page 143 - Textos de Matemática Vol. 46

SOME REMARKS ON PULLBACKS IN GUMM CATEGORIES 133
so that the rectangle formed by the back and right faces is a pullback. We can apply Proposition 4.1 to conclude that both the back and right faces above are pullbacks. By the Barr-Kock Theorem the right face of diagram (5.1) is then a pullback, hence so is the front face of (5.1). ⇤
As a consequence of Proposition 5.1 we give a new proof of Theorem 4.8 of [15] stating that every central and split extension is a trivial extension for the Galois structure associated with any Birkho↵ subcategory of an exact Goursat category. Let us briefly recall the main definitions, and we refer to [15] for more details.
When C is an exact category and X a full replete subcategory of C
I
one calls X a Birkho↵ subcategory of C when X is stable in C under subobjects and regular quotients. Equivalently, all X-components ⌘X of the unit ⌘: 1C ) I of the adjunction are regular epimorphisms (the right adjoint is assumed to be a full inclusion and will not be mentioned explicitly), and the naturality square
//
Coo ? ?_X, (5.2)
X ⌘X ////IX
f If
✏✏✏✏ // // ✏✏✏✏ Y⌘Y IY
(5.3)
is a pushout for any regular epimorphism f : X // // Y .
A regular epimorphism f : X // // Y is called a trivial extension when the
naturality square (5.3) is a pullback. It is called a central extension when it is “locally” trivial: there exists a regular epimorphism y : Z // // Y such that the pullback of f along y is a trivial extension.
Theorem 5.2. (Theorem 4.8 of [15]) Let C be an exact Goursat category, and X a Birkho↵ subcategory of C. Then every central and split extension is necessarily a trivial extension.
Proof. Let f : X // // Y be both a central extension and a split epimorphism.
By definition, there exists a regular epimorphism y : Z // // Y such that the pullback of f along y is a trivial extension. So, in the following commutative