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Cocategories over Commutative Algebras: Hopf Algebroids 431

(2) (unit)

e×X
X = {∗} × X −−−−→ G × X


α
X.

(2.9) Remark. The related groupoid X"G#(∗) is defined by X"G#(0) = X and
X"G#(1) = G × X with structure morphisms

e = e G × X : X"G#(0) → X"G#(1) = G × X,
l = pr 2 : X"G#(1) = G × X → X = X"G#(0) and
r = α : X"G#(1) → X = X"G#(0).
For the composition we need the natural isomorphism

θ : X"G#(1) × X"G#(1) → G × G × X
r X(G)(0) l

given by pr 1 θ = pr G pr 1 , pr 2 θ = p G pr 2 ,and pr 3 θ = p X pr 1 . Then composition
m is defined by the following commutative diagram

θ
X"G#(1) × X"G#(1) −−−−→ G × G × X
r X(G)(0) l
 
µ×X
 
m
X"G#(1) ←−−→ G × X.
The unit and associativity properties of m come from the unit and associativity prop-
erties of µ : G × G → G.

(2.10) Definition. With the above notation X"G#(∗) is the groupoid associated to
the G-action on the object X in C. It is also called the translation category.

§3. Cocategories over Commutative Algebras: Hopf Algebroids

(3.1) Notation. Let (c\alg/R) be the category of commutative algebras over a com-
mutative ring R. The coproduct in this category is the tensor product over R and the

initial object is R. Let g : A → A and f : A → A be two morphisms in (c\alg/R).
q q

The cofibre coproduct in the category (c\alg/R) is A −→ A ⊗ A A ←− A where

q (x ) = x ⊗1and q (x ) = 1⊗ x . The tensor product over A defining the cofibre

coproduct is formed with the right A-module structure x a = x g(a) and the left

A-module structure ax = f (a)x for a ∈ A, x ∈ A ,and x ∈ A .