Algebraic Combinatorics I: Association Schemes (Mathematics by Eiichi Bannai PDF

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Extra info for Algebraic Combinatorics I: Association Schemes (Mathematics lecture note series) (Bk. 1)

Example text

Counit-cylinder µ as α = B(f, 1)µ for a unique f : B An indexed colimit in B is nothing but an indexed limit in Bop ; but it is usual to replace /G B, but now have F : Kop /G V. 5) = [Kop , V] F, B(G−, B) , with unit ν: F /G B(G−, F ⋆ G). 7) either side existing if the other does. 5) can be written B(B, {F, G}) ∼ = {F, B(B, G−)}, B(F ⋆ G, B) ∼ = {F, B(G−, B)}. 2, in terms of ordinary limits in V0 . 5) F ⋆G∼ =G⋆F for F : Kop /G V and G : K /G V. 33) immediately gives the values of {F, G} and /G B; namely F ⋆ G when F is representable, for any G : K {K(K, −), G} ∼ = GK, K(−, K) ⋆ G ∼ = GK.

Clearly a fully-faithful T : A /G B has one, and if and only if the inclusion A′′ /G B has one. A′ /G B has a left A full subcategory A of B is called reflective if the inclusion T : A adjoint. This implies, of course, that A0 is reflective in B0 , but is in general strictly stronger. It follows – since it is trivially true for ordinary categories – that every retract (in B0 ) of an object of the reflective A lies in the repletion of A. /G A that ϵ : ST /G 1 is When A is reflective we may so choose the left adjoint S : B 2 /G R = T S : B /G B satisfies R = R, ηR = Rη = 1.

Moreover V0′ admits all limits and colimits which are small by /G V ′ preserves all limits and colimits that exist reference to Set′ ; and the inclusion V0 0 in V0 . ) Then, for V-categories A and B, we always have a V ′ -category [A, B], where Set′ /G V ′ preserves limits, to say that is taken large enough to include ob A. 2 – that is, as a V-category – is to say that [A, B](T, S) ∈ V for all T , S. That a similar thing is possible for a perfectly general V follows from Day [17]. Using his construction, one can show that there is always a suitable V ′ whose underlying ordinary category V0′ is a full reflective subcategory of [V0op , Set′ ], into which V0 is embedded by Yoneda.

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