EXERCISES IN CATEGORY THEORY 6
1. Limits
(1) Let J be the category below:
0
i

X
p0
//
p1

D(0)
D(i)

1
j
// 2 D(1)
D(j)
// D(2)
and consider a diagram D : J → C. Show that a cone (X, p) on D is the same thing
as a pair of arrows X → D(0) and X → D(1) making the square above commute.
Show therefore that the limit of D is exactly the pullback of D(i) and D(j).
(2) Let C be a category with products. Show that each representable functor C(X, −) :
C → Set preserves products.
(3) Let J and C be categories. Given X ∈ C the constant functor ∆X : J → C at X is
defined by ∆X(j) = X for all j ∈ J and sends all morphisms of J to the identity on
X. Given D : J → C show that a natural transformation p : ∆X → D is the same
thing as a cone (X, p) on D.
(4) Show that each morphism f : X → Y ∈ C determines a natural transformation
between constant functors ∆f : ∆X → ∆Y and observe that a morphism of cones
(X, p) → (Y, q) over D amounts to an arrow f such that the triangle of natural
transformations
∆X
p
##GGGGGGGGGG
∆(f)
// ∆(Y )
q

D
(5) Given a functor F : A → B and object X ∈ B the comma category F/X has objects:
triples (A, p : FA → X) and morphisms f : (A, p : FA → X) → (B, q : FB → X) are
arrows f : A → B such that q ◦ Ff = p.
Show that constant functors themselves define a functor ∆ : C → [J, C] and that
Cone(D) is the comma category ∆/D.
(6) Given a diagram D : J → Set show that the limit of D is given by the set of cones
Cone(1, D) over D with base the 1-element set.
Date: October 23, 2014.
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