Basic Category Theory Tom Leinster
Chapter 1

Exercise 3.26
A natural transformation is an isomorphism precisely when each component is an isomorphism

Exercise 3.27
The functor categories and are isomorphic

Exercise 3.3
For a group considered as a category, equivalence classes of functors under natural isomorphism correspond to conjugacy classes of elements of

Exercise 3.31
The set of permutations and the set of total orders on a given set are isomorphic, but not naturally

Exercise 3.32
functor is an equivalence of cateogories if and only if it is full, faithful and essentially surjective on objects

Exercise 3.34
Equivalence of categories is an equivalence relation
Chapter 2

Exercise 1.15
Left adjoints preserve initial objects

Exercise 1.16
An exploration of “interesting” functors between and the category of left sets

Exercise 1.17
Exhibiting a chain of five adjoint functors between and the category of presheaves on a topological space

Exercise 2.11
Every adjunction restrictions to an equivalence between (full) subcategories

Exercise 2.13
Formulating quantifiers as functors

Exercise 3.1
An adjunction where the unit and counit are isomorphisms is an equivalence

Exercise 3.11
If a functor has a left adjoint, and there is at least one set with having more than one element, then every component of the counit is an injective map of sets

Exercise 3.12
The category of sets with partial functions is equivalent to the category of pointed sets
Chapter 4

Exercise 1.27
implies (The Yoneda embedding is injective on isomorphism classes of objects)

Exercise 1.28
Finding a representation of a certain functor

Exercise 1.29
Finding a representation of the forgetful functor from the category of commutative rings

Exercise 1.31
Finding a representation of the functor sending a category to its set of morphisms
Chapter 5

Exercise 1.34
Characterizing equalizers in terms of pullbacks

Exercise 1.35
The pullback pasting lemma

Exercise 1.36
Limit cones over a diagram are in correspondence with maps into the limit object

Exercise 1.38
A category with products and equalizers has all limits

Exercise 1.4
Subobjects (monics) in are isomorphic when they have the same image

Exercise 1.41
Characterizing monic arrows in terms of a pullback square

Exercise 1.42
A pullback of a monic map is monic

Exercise 2.23
Characterizing maps that are epic but not surjective in some categories

Exercise 2.24
Two epics are isomorphic in when they induce the same equivalence relation

Exercise 2.25
Split monics are regular monics are monics

Exercise 2.26
The inclusion in is monic and epic but not an isomorphism. However, a map is an isomorphisms when it is monic and regular epic
Chapter 6

Exercise 1.6
Interpreting a chapter theorem about a right adjoint to the limit functor

Exercise 1.21
If are presheaves with , then either or is trivial

Exercise 2.2
In appropriate categories, a natural transformation in a functor category is monic precisely when each component is monic

Exercise 2.24
The slice of a presheaf category is equivalent to presheaves on the category of elements
Other

Construction of Kan Extensions
We define Kan extensions, then give an explicit construction of the Kan extension through colimits and prove that our construction is actually the Kan extension.

Yoneda Lemma
Proof of the Yoneda lemma