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430 Appendix III: Elliptic Curves and Topological Modular Forms

(2.2) Example. In the category of sets a category object G(∗) is a groupoid when
every morphism u ∈ G(1), which is defined u : l(u) → r(u), is an isomorphism,
and in this case the morphism i is the inverse given by i(u) = u −1 : r(1) → l(u).

(2.3) Remark. If i and are two groupoid structures on a category (C(0), C(1), l,

r, e, m) then i = i . To see this, we calculate as with groups using the associative
law

i (u) = m(i (u), m(u, i (u))) = m(m(i (u), u), i (u)) = i (u).

This means that a groupoid is not a category with an additional structure, but a cat-
egory satisfying an axiom, namely i exists. Also we have ii = G(1), the identity on
G(1) by the same argument.
In general we can think of groupoids as categories where every morphism is
an isomorphism, and the process of associating to an isomorphism its inverse is an
isomorphism of the category to its opposite category which is an involution when the
double opposite is identified with the original category.
(2.4) Example. A groupoid G(∗) with G(0) the final object in C is just a group
object in the category C.

(2.5) Definition. A morphism u(∗) : G (∗) → G (∗) of groupoids in C is a mor-
phism of categories in C.

(2.6) Remark. A morphism of groupoids has the additional groperty that i u(1) =

u(1)i . This is seen as with groups from the relation m (i u(1), u(1)) = l e =

m (u(1)i , u(1)). We derive (2.3) by applying this to the identity functor.
(2.7) Definition. Let grpoid(C) denote the full subcategory of cat(C) determined by
the groupoids.
There is a full subcategory grp(C) of grpoid(C) consisting of those groupoids
G(∗) where G(0) is the final object in C. This is the category of groups over the
category C.
The category grpoid(set) of groupoids over the category of sets is the just the
category of small categories with the property that all morphisms are isomorphisms.
Also grp(set) is just the category of groups (grp) of sets.
(2.8) Example. Let G be a group object in a category C with fibre products. An
action of G on an object X of C is a morphism α : G × X → X satisfy two axioms
given by commutative diagrams
(1) (associativity)

G×α
G × G × X −−−−→ G × X
 
α
 
µ×X
α
G × X −−−−→ X
where µ : G × G → G is the product on the group object G,and