#### Basic Category TheoryTom 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 $[\mathbf{A}^{\text{op}}, \mathbf{B}^{\text{op}}]$ and $[\mathbf{A}, \mathbf{B}]^{\text{op}}$ are isomorphic
• Exercise 3.3 For a group $G$ considered as a category, equivalence classes of functors $\mathbb{Z} \rightarrow G$ under natural isomorphism correspond to conjugacy classes of elements of $G$
• 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 $\mathbf{Set}$ and the category of left $G$-sets
• Exercise 1.17 Exhibiting a chain of five adjoint functors between $\mathbf{Set}$ 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 $\mathbf{A} \rightarrow \mathbf{Set}$ has a left adjoint, and there is at least one set $A$ with $U(A)$ 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 $\Hom(\dash, A) \cong \Hom(\dash, A')$ implies $A \cong A'$ (The Yoneda embedding is injective on isomorphism classes of objects)
• Exercise 1.28 Finding a representation of a certain functor $\mathbf{Grp} \rightarrow \mathbf{Set}$
• 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 $\mathbf{Set}$ 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 $\mathbf{Set}$ when they induce the same equivalence relation
• Exercise 2.25 Split monics are regular monics are monics
• Exercise 2.26 The inclusion $\mathbb{Z} \rightarrow \mathbf{Q}$ in $\mathbf{Rng}$ 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 $X, Y$ are presheaves with $\Hom(\dash, A) \cong X + Y$, then either $X$ or $Y$ 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