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432 Appendix III: Elliptic Curves and Topological Modular Forms

(3.2) Definition. A cocategory object in this category (C alg R) is a sextuple
op op op op
(A, ,η L ,η H ,ε, ) such that the sextuple (A, ,η ,η ,ε , ) is a category
L
R
object in the dual category (C alg R). op
In particular the category axioms (Cat1)–(Cat4) will correspond to commutative
diagrams of commutative algebras.
(3.3) Remark. In particular the objects A and in the cocategory object (A, ,η L ,
η H ,ε, ) are commutative algebras over R. The first two structure morphisms
η L ,η R : A → define a left and right A-module structure on by xa = xη R (a) and
ax = η L (a)x in for a ∈ A and x ∈ . The third structure morphism ε : → A
satisfies the augmentation relation ε L (a) = a and ε R (a) = a for a ∈ A. In particular
the following diagram is commutative

η L η R
A −−−−→ ←−−−− A

ε

A

where the diagonal morphisms are identities. This is the dual to (Cat1).
The cocategory cocomposition, called the comultiplication on , : → ⊗ A
q q
is a morphism of A-bimodules. Using the morphisms −→ ⊗ A ←− ,wesee

that the composites q η L : A → ⊗ A and q η R : A → ⊗ A defineleftand

right A-module structures on ⊗ A .

In terms of elements z ∈ we can write (z) = z ⊗ z and for a ∈ A the
i i i
left and right linearity has the form

(az) = a (z) (az ) ⊗ z and (za) = (z)a = z ⊗ (z a).
i i i i
i i
The opposite to (Cat3) is the associativity of comultiplication which satisfies the
following commutative diagram

−−−−→ ⊗ A
 
 
⊗

⊗
⊗ A −−−−→ ⊗ A ⊗ A
The dual to (Cat4) is counit, called the counit ε of the Hopf algebroid which
satisfies the following commutative diagram




A ⊗ A ←−−−− ⊗ A −−−−→ ⊗ A A
ε⊗ A ⊗ A ε