By Joseph J. Rotman

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**Extra info for An Introduction to the Theory of Groups, 4th Edition**

**Example text**

2, E). This gives a category covering each object of B. We would like to have functors covering each morphism of B in either a contravariant or covariant "pseudo-functorial" way. To do this, let J B : CB -+ C denote the inclusion functor. , "Pf: J B, -+ JB f*) satisfying two axioms. 1. , cp" 0 ("Pf}E' = cp). Pf) E' E' ,.. '! , ("Pf}E) when it is clear which component of the natural transformation is relevant. , f*E' represents the functor C (E', - }f: C B' -+ Yet-o), where the subscript (f) indicates morphisms cp with P(cp) = f.

J -+ C. A cleavage is normal if and only if the corresponding function satisfies y(l) = 1 and it is split if and only if y is also a group homomorphism. This accounts for the term split and shows that a functor which admits a cleavage need not admit a split cleavage. § 2. 1. J) denote the image of in by the forgetful functor. J. J which are cartesian for some choice of cleavages. J) together with a number of properties of fibrations which follow easily from these characterizations.

1 Proof. The "only if" is obvious. Conversely, suppose for each T B , there is an SB = lari TB. Then under the adjointness relation - - (SB' f* E, f* SBE) (f* E, T B, f* SBE) a* E, 1* TB SB E) (f* E, i* E) there is a unique r/J: SB' f* f* SB with TTiI = 11*. 6 to determine a functor S: iff iff. By construction PS = P and TS = id. Adjointness is easily checked directly. We now wish to discuss some adjoint functors between [/jzti/o(gj), and Cfia/91. Suppose F: d gj is a functor.