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Groupoids in a Category 429

(1.5) Example. A category object C(∗) with C(0) the final object in C is just a
monoidal object C(1) in the category C.
(1.6) Example. The pair of morphisms consisting of the identities u(0) = C(0) and
u(1) = C(1) is a morphism u(∗) : C(∗) → C(∗) of a category object called the

identity morphism on C(∗).If u(∗) : C (∗) → C (∗) and v(∗) : C (∗) → C(∗) are

two morphisms of category objects in C, then (vu)(∗) : C (∗) → C(∗) defined by
(vu)(0) = v(0)u(0) and (vu)(1) = v(1)u(1) is a morphism of category objects in C.
(1.7) Definition. With the identity morphisms and the composition of morphisms in
(1.5) we see that the category objects and the morphisms of category objects form
a category called cat(C). There is a full subategory mon(C)ofcat(C) consisting of
those categories C(∗) where C(∗) is the final object in C. This is the category of
monoids over the category C.
(1.8) Remark. The category cat(set) of category objects over the category of sets is
the just the category of small categories where morphisms of categories are functors
between the categories and composition is composition of functors. Also mon(set) is
just the category of monoids. There is an additional structure of equivalence between
morphisms as natural transform of functors, and this leads to the notion of 2-category
which we will not go into here.

§2. Groupoids in a Category

In the case of a groupoid where each f is an isomorphism with inverse i( f ) = f −1 ,
this formula defines a map i : C(1) → C(1) with domain and range interchanged
r(i( f )) = il( f ) and l(i( f )) = r( f ). There is also the inverse property m( f, i( f )) =
e(l( f )) and m(i( f ), f ) = e(r( f )). In the next sections we have the axioms.
(2.1) Definition. Let C be a category with fibre products. A groupoid G(∗) in C is
a septuple (G(0), G(1), l, r, e, m, i) where the sextuple (G(0), G(1), l, r, e, m) is a
category object and inverse morphism i : G(1) → G(1) satisfying in addition to
axioms (cat1)–(cat4) the following axioms:

(grpoid1) The following compositions hold li = r and ri = l.
(grpoid2) The following commutative diagrams give the inverse property
of i : G(1) → G(1)

(G(1),i) (i,G(1))
G(1) −−−−→ G(1) × G(1) G(1) −−−−→ G(1) × G(1)
r G(0) l r G(0) l
   
   
m m
l r
e e
G(0) −−−−→ G(1) G(0) −−−−→ G(1).
Here the same symbol is used for an object and the identity on an object.