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Extra resources for Algebra, Topology, and Category Theory. A Collection of Papers in Honor of Samuel Eilenberg
In Case 2 it is true because KC is a free ideal ringoid (that is, every subfunctor of a representable is a free KCmodule). In Case 3, at least when C is finitely generated, the conjecture is Serre's. REFERENCES         N . Bourbaki, Eléments de m a t h é m a t i q u e s ; Algèbre c o m m u t a t i v e . Actualités Sei. Indust. 1290, 1293, H e r m a n n , Paris, 1961. P. M. C o h n , Free ideal rings. J. Algebra 1 (1964), 47-69. R. Diaconescu, Thesis, Dalhousie University, 1973.
Hence IM is free. 5. If U : C - • Ο takes all morphisms to monomorphisms, c then the right adjoint to A b ° - • A b is exact if and only if the components op of (D, U) have right zeros for all D e | Ο |. Proof. The right adjoint composed with the evaluation functor at D is given by homing with ZD(Z), [/_). Hence the right adjoint is exact if and op only if the components of (Y, D(D, l / _ ) ) have right zeros for all D. )) = (D, U). 1, where we used the fact that if Ν is a group and I is projective, then l\l is trivial.
F o r example, the linear tree of Fig. 7 is satisfied by 0 - > A if and only if A has binary products. We will call such properties diagrammatic properties. V 3! 3 Fig. 7 Linear trees d o not suffice. F o r example, the property that A is linearly connected requires a nonlinear tree such as that shown in Fig. 8. ) V V 3 • <— · Fig. 8 Fig. 9 Fig. 10 PROPERTIES INVARIANT W I T H I N 3 V I V 59 EQUIVALENCE TYPES OF CATEGORIES I I I 0 3 V in V 1 • • I m m • · 1 • —• · • —• · I \/ 3 V 1 • —• · • • —• · 1 I in ι ι • 0 · Fig.