SAMARAN "et.al."

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triple (A,t,c) where A is an object in с ^ : A ^^ AgiA (comultiplication) and c:A ^ К (co-unit) are /3,-morphisms such that the following diagrams commute:

- ^A®A

A8>A .

|1д®Ф

RgiA

- Wte>Ag>A

>A8>K

* ®^A

Equivalently , a <»,-cogebra is a e^^-algebra where {cf^^,<S>) is a category with multiplication in the obvious way. Morphisms, ^-products, etc., of cogebras are similarly fined in the obvious dual fashion. In particular, if T о iif = Ф we shall say that (A,i>,c) is co-commutative The (3,-cogebras and their morphisms constitute a category Co(c„(8>).

DEFINITION 1.4. ([4], Part I, Sec. 5). A ^.-bigebra is an

object A of (C-^ig)) such that

( i ) (А,7Г, I) is a (З-algebra,

( ii ) (A,f ,c) is a /^-cogebra,

( iii ) c:A -♦ К is a morphism of C^-algebras,

( iv ) \!r:A •♦ ША is a morphism of ^--algebras.

It is a trivial fact that (iii) and (iv) can be placed by

( iii ) ' I:K 4 A is a morphism of (З-cogebras, (iv)' 7г:А®А -♦A is a morphism of c^-cogebras.

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