By W Casselman, Armand Borel, W. Casselman
This was once the convention on $L$-functions and automorphic types. the 2 volumes are actually classics.
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Extra resources for Automorphic Forms, Representations, and L-Functions (Proceedings of Symposia in Pure Mathematics)
There are enough strong cartesian morphisms. Proof. "i. " and "iii. " are obvious. "ii. " is a simple exercise and "iv. " is shown by choosing a strong cartesian morphism for each f: B' --J> B in P,B and each E E iffJiJ. The domains of these morphisms determine functions f*: ob iffB --J> ob iffB, and the morphisms themselves are the components of the desired transformations. The universal mapping properties imply that there is a unique way to extend these functions to functors for which the transformations are natural.
5 is split. jj 54 J. W. J8, F'): (g: B ---*F(A))---* ---* (g: B ---* F' (T (A))): (f, h) ---* (f, T (h)) . ((Of)g) = (Of)g so T. not only is cartesian but preserves the canonical split cleavages. J8). Furthermore, if T and T' differ by a natural equivalence t: T ---* T' with F't = Ip then T. and T; differ by a natural equivalence t. J8D LlF = F. J8, -) . J8 and the second shows that it is a special case of lifting a generalized homotopy, since P 2 = lali Qp. Hence there is a "best possible" H making everything commute and we must show that "best possible" is the same as cartesian in this circumstance.
3. If, furthermore, the square with the vertical arrows from top to bottom is a pullback, then 'I} is a natural equivalence; since, by the preceding result there is a T1 rari 8 such that 'l}1: GT1 -+ QF is the identity natural transformation, and T differs from T1 by a natural equivalence. --c= 'I} = 0 GT QF'IjJ' : QF -+ GT . a') If Q = lali P then 0 = id so this reduces to 'I} = QF'IjJ': QF-+GT. b') If, furthermore, T = lali, 8, then 'l}8 = la. c') If, furthermore, the square with the vertical arrows from top to bottom is a pushout, then 'I} is a natural equivalence.
Automorphic Forms, Representations, and L-Functions (Proceedings of Symposia in Pure Mathematics) by W Casselman, Armand Borel, W. Casselman