Basic Category Theory

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

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

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

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

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

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