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Categories in a Category 427

I wish to thank Tilman Bauer, Michael Joachim, and Stefan Schwede for the help
and encouragement in preparing this appendix.

§1. Categories in a Category

We begin by a short introduction to the concept of category object and consider the
axiomatic framework of categories in a category.
(1.1) Small Categories as Pairs of Sets. Let C be a small category which means
that the class of objects C(0) is a set. Form the set C(1) equal to the disjoint union of
all Hom(X, Y) for X, Y ∈ C(0). These two sets are connected by several functions.
Firstly, we have the domain (left) and range (right) functions l, r : C(1) → C(0)
defined by the requirement that

l(Hom(X, Y)) ={X} and r(Hom(X, Y)) ={Y}
on the disjoint union. For f ∈ C(1) we have f : l( f ) → r( f ) is a notation for the
morphism f in C.
Secondly, we have an identity morphism for each object of C which is a function
e : C(0) → C(1) having the property that le and re is the identity on C(0).
Thirdly, we have composition gf of the two morphisms f and g, but only
in the case where r( f ) = l(g). Hence composition is not defined in general
on the entire product C(1) × C(1), but it is defined on all subsets of the form
Hom(X, Y) × Hom(Y, Z) ⊂ C(1) × C(1). This subset is called the fibre product
of r : C(1) → C(0) and l : C(1) → C(0) consisting of pairs ( f, g) ∈ C(1) × C(1)
where r( f ) = l(g). The fiber product is denoted C(1) × C(1) with two projec-
r C(0) l
tions r, l : C(1) × C(1) → C(0) defined by
r C(0) l

l( f, g) = l( f ) and r( f, g) = r(g).
Then composition is defined m : C(1) × C(1) → C(1) satisfying lm( f, g) =
r C(0) l
l( f, g) = l( f ) and rm( f, g) = r( f, g) = r(g). Now the reader can supply the unit
and associativity axioms.
op
Fourthly, the notion of opposite category C where f : X → Y in C be-
g
comes f op : Y → X in C op and (gf ) op = f op op can be described as C op =
(C(0), C(1), e, l op = r, r op = l, m op = mτ) where τ is the flipin the fibre product
τ : C(1) × C(1) → C(1) × C(1).
r C(0) l l C(0) r

(1.2) Definition. Let C be a category with fibre products. A category object C(∗)
in C is a sextuple (C(0), C(1), l, r, e, m) consisting two objects C(0) and C(1) and
four morphisms
(1) l, r : C(1) → C(0) called domain (left) and range (right),