By S. A. Amitsur, D. J. Saltman, George B. Seligman

ISBN-10: 082185013X

ISBN-13: 9780821850138

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**Extra resources for Algebraists' Homage: Papers in Ring Theory and Related Topics**

**Example text**

7 c. Proof. We have T(Of) = Of 0 'YJf. 3, T(Of) is cartesian if and only if 'YJf is an equivalence. To translate these characterizations of [SGA] into forms more reminiscent of topology, we consider a number of properties, all of which turn out to be equivalent. 8. Let P: iff --J> PJ, P: i --J> P,B and T: iff --J> i with PT=P. i. 2, E). - If T: E -+E- satisfies PT - for E E = - P then PT(E) TE = PE. IP(E) However, T need not commute with QE and QT(E)' but the adjunction natural transformation ip: 1-+ QT(E) PT(E) gives rise to a transfer natural transformation 'YJE = if TE QE : TE QE -+ QT(E) - Q P TQ = Q PQ = Q, (since omitting subscripts.

1tJ --+ QP. The bottom square, the back square, the two right hand squares and the two front squares all commute. Furthermore the front square with vertical arrows from top to bottom is a pullback since (-)" has a left adjoint. Hence there is a unique "ij: d --+ d" such that G""ij = 1p G and S""ij = LIS. This last equation is condition c) of those to be verified. For the others we argue as follows: a) By the commutativity relations and the properties of tp, we have GdDij = G and SdD"ij = S, so dD"ij = lsd' b) Similarly, GdR"ij c) Finally, G" ij T = G" T" LI and = GTS and SdRij = STS so dRij = TS.

33 Fibred and Cofibred Categories We would like to extend this equivalence between the notion of functors with cleavages and the notion of pseudo-functors to an equivalence of categories. To do so, consider a commutative diagram T E offunctors, where P and respectively. IJ and if f: B' -+ B then there is a unique natural transfor'fJt: T B, 0 f* -+ f* 0 TB such that P ('lit) = i B, and 010 'YJI = T (0 1). 6. g I for B" -+ B' -+ B or CI,g 0 (g* * 'YJI) 0 ('YJg * f*)) = * 'YJlg 0 (T CI,g)· 1. 7.

### Algebraists' Homage: Papers in Ring Theory and Related Topics by S. A. Amitsur, D. J. Saltman, George B. Seligman

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