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Cocategories over Commutative Algebras: Hopf Algebroids 433

where diagonal morphisms are natural isomorphisms. A cocategory (A, ) is a
cogroupoid provided there is a fifth structure morphism c : → satisfying the
additional commutative diagrams

c⊗ ⊗c
⊗ A −−−−→ ⊗ A ⊗ A −−−−→ ⊗ A
%  % 
(4)  φ( )  φ( )


 
εη L εη R
−−−−→ −−−−→ .
Another name for such a cocategory or cogroupoid object (A, ) is Hopf alge-
broid. We also speak of as a Hopf algebroid over A. Usually we would require that
is flat over A in order that the categories of comodules are abelian.
(3.4) Remark. We have defined a cocategory object in the category (c\alg/R),
however the same definition applies in any category C with finite colimits, in par-
ticular, an initial object is used and the cofibre coproduct construction.
A single cocategory object in a category C produce a category for each object in
C.

(3.5) Remark. Let (A, ) be a cocategory object in a category C, and let Z be any
object in C. Then the cocategory object defines a category (in the category of sets)
with objects Hom C (A, Z) and morphism set Hom C ( , Z).
(1) The function assigning to each object its identity is

e = Hom C (ε, Z) :Hom C (A, Z) → Hom C ( , Z).
(2) The domain and range morphisms are given by

l = Hom C (η L , Z) :Hom C ( , Z) → Hom C (A, Z) and
r = Hom C (η R , Z) :Hom C ( , Z) → Hom C (A, Z).
(3) composition morphism in the category corresponding to the object Z on the
morphism sets Hom C ( , Z) is the inverse of the following isomorphism used to
define the cofibre coproduct
Hom C ( , A) × Hom C ( , Z) ← Hom C ( ⊗ A , A),
r Hom C (A,Z) l
and the morphism Hom C ( , Z) :Hom C ( ⊗ A , Z) → Hom C ( , Z) induced
by .
In the case where (A, ) is a cogroupoid with additional structure morphism
c : → the resulting category with morphisms Hom C ( , Z) is a groupoid
with the inverse map
i :Hom C ( , Z) → Hom C ( , Z)

given by i( f ) = fc.