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428 Appendix III: Elliptic Curves and Topological Modular Forms

(2) e : C(0) → C(1) a unit morphism, and
(3) m : C(1) × C(1) → C(1) called multiplication or composition satisfying
r C(0) l
the following axioms:
(Cat 1) The compositions le and re are the identities C(0).
(Cat 2) Domain and range are compatible with multiplication.
m m
C(1) × C(1) −−−−→ C(1) C(1) × C(1) −−−−→ C(1)
r C(0) l r C(0) l
 
 
   
l pr 2 r
pr 1
l r
C(1) −−−−→ C(0) C(1) −−−−→ C(0)
(Cat 3) (associativity) The following diagram is commutative
m×C(1)
C(1) × C(1) × C(1) −−−−→ C(1) × C(1)
r C(0) l r C(0) l r C(0) l
 
m
 
C(1)×m
m
C(1) × C(1) −−−−→ C(1)
r C(0) l
(Cat 4) (unit property of e) m(C(1), er) and m(el, C(1)) are each the identities
on C(1).

(1.3) Definition. A morphism u(∗) : C (∗) → C (∗) from the category object

C (∗) in C to the category object C (∗) in C is a pair of morphisms u(0) : C (0) →

C (0) and u(1) : C (1) → C (1) commuting with the four structure morphisms of

C (∗) and C (∗). The following diagrams are commutative

e l r

C (0) −−−−→ C (1) C (1) −−−−→ C (0) C (1) −−−−→ C (0)
     
     
u(1) u(0) u(0) u(1) u(0)
u(0)
e l r

C (0) −−−−→ C (1) C (1) −−−−→ C (0) C (1) −−−−→ C (0)
and
m
C (1) × C (1) −−−−→ C (1)

r l
C (0)
 

u(1) × u(1)  u(1)
u(0)
m

C (1) × C (1) −−−−→ C (1).
r l
C (0)
(1.4) Example. A category C(∗) in the category of sets (set) is a small category as
in (1.1).