By Boyer Ch. P.
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Extra info for 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients
51. Are group objects domain-type or target-type gadgets? Suppose G is a group object in C. What conditions must you impose on a functor F : C -+ D in order to conclude that F(G) E D is also a group object? Let's check that these things are correctly named. 52. (a) Check that, in the category of pointed sets, a group object is just an ordinary group. (b) Show that a group G E is a group object if and only if G is abelian. (c) Write GL,, (R) to denote the set of all n x n invertible matrices. It is a subset of Rn2 , so we can give it the subspace topology.
Let's start with functors R : C Taking X = L(Y), we have a natural isomorphism (D : morv(L(Y), L(Y)) > morc(Y, RL(Y)). Applying this to the identity idL(Y) gives us a map a : Y -+ RL(Y). 74. Show that there is a commutative diagram more (X, Y) morc(X, Y) L a* more (L(X) I L(Y) ) morc(X, RL(Y)). Thus, the effect of L on morphisms can be identified with the map Q*, which is defined by composition of morphisms. 75. 74 shows that the maps L and a* are equivalent maps. 2 we defined what equivalence means in categorical terms-what category are we working in here?
1 that we can, without causing any trouble, take any diagram and augment it by including all composite arrows and identity arrows that are not already present in the diagram, and the result is a category. If we apply this construction to the shape diagram -3 * +- o, we obtain a category Z. The key observation is that the diagram Xf )Y( 9Z determines a functor F:I 29 2. Limits and Colimits 30 given explicitly by F(*) = F(o) = F( - *) = f Y, F(* - o) = g, Z, and of course F carries identities to identities.