Category Theory

A category \(\mathcal C\) consists of

• a set of objects \(\mathrm{ob}(\mathcal C)\)
• sets of morphisms \(\mathrm{Hom}_{\mathcal{C}}(A,B)\) called homsets between each pair of objects \(A,B\) in \(\mathcal{C}\)
• binary operation \(\circ\colon \mathrm{Hom}_{\mathcal{C}}(B, C)\times\mathrm{Hom}_{\mathcal{C}}(A, B)\to \mathrm{Hom}_{\mathcal{C}}(A, C)\) called composition of morphisms

satifying the following properties:

• Identity: \(\forall\,X \in \mathcal{C},\exists\, \mathrm{id}_{X}:X\to X\), such that for \(f\in \mathrm{Hom}_{\mathcal{C}}(A,X)\) and \(g\in \mathrm{Hom}_{\mathcal{C}}(X,A)\), \(f\circ \mathrm{id}_{X}=f\) and \(\mathrm{id}_{X}\circ g= g\) holds.
• Associativity: If \(f\in \mathrm{Hom}(A, B), g\in\mathrm{Hom}(B, C), h\in\mathrm{Hom}(C, D)\), then \(h\circ(g\circ f) = (h\circ g)\circ f.\)

A category whose only morphisms are the identity morphisms: \[ \mathrm{Hom}_{\mathcal{C}}(X, Y) = \begin{cases} \{\mathrm{id}_X\} & \text{if $X=Y$,}\\ \varnothing & \text{if $X\neq Y$.} \end{cases} \]

The class of functors from \(\mathcal{C}\) to \(\mathcal{D}\) with morphisms being the natural transformations. It is commonly written as \( \mathcal{D}^\mathcal{C} \).

• Dual Category

An opposite category \( \mathcal{C}^{\mathrm{op}} \) is a category where all the arrows are reversed from a given category \( \mathcal{C} \).

• Posetal Category, (0,1)-Category

A category is called thin, if there are at most one morphism in each homset.

This can effectively represent the sturcture of a poset (partially ordered set). One example of this is the category of open sets \( \mathrm{Op}(X) \) of a topological space \( X \), in which the inclusions \( \iota\colon V \hookrightarrow U \) are the morphisms.

• Both monomorphism and retraction \(\iff\) Both epimorphism and section \(\iff\) isomorphism.

• \[ \ker f := \{x\in X : f(x) = 1_Y\} \]

• \[ \operatorname{coker} f := Y/\operatorname{im} f \]

• \[ \operatorname{im} f := \{y \in Y : y = f(x)\} \]

• \[ \operatorname{coim}f := X/\ker f \]

A functor from an object to the homset associated to that object.

Covariant Functor \( \mathrm{Hom}(A, -)\colon \mathcal{C} \to \mathbf{Set} \):

• \( \mathrm{Hom}(A,X) \) is the set of morphisms from \( A \) to \( X \).
• \( \mathrm{Hom}(A,f)\colon \mathrm{Hom}(A,X) \to \mathrm{Hom}(A,Y) \) with \( f\colon X\to Y \) is a morphism given by \( g\mapsto f\circ g \).

Contravariant Functor \( \mathrm{Hom}(-, B)\colon \mathcal{C} \to \mathbf{Set} \):

• \( \mathrm{Hom}(X,B) \) is the set of morphisms from \( X \) to \( B \).
• \( \mathrm{Hom}(h,B)\colon \mathrm{Hom}(Y,B) \to \mathrm{Hom}(X,B) \) with \( f\colon X\to Y \) is a morphism given by \( g\mapsto g\circ h \).

Additionally, \( \mathrm{Hom}(-,-)\colon \mathcal{C}^{\mathrm{op}}\times \mathcal{C} \to \mathbf{Set} \) is a bifunctor.

A diagonal functor \( \Delta\colon \mathcal{C} \to \mathcal{C}^{\mathcal{J}} \) is a functor from a category \( \mathcal{C} \) to the functor category \( \mathcal{C}^{\mathcal{J}} \) where \( \mathcal{J} \) is a small index category, that maps each object \( A \in \mathcal{C} \) to the constant functor \( \Delta(A)\colon \mathcal{J} \to \mathcal{C}\colon N \mapsto A\colon (f\colon N\to M) \mapsto \mathrm{id}_A\).

The name diagonal arises as the generalization from the simple case where \( \mathcal{J} \) has only two objects, in which the constant functor is effectively the same to the diagonal functor \( \Delta\colon \mathcal{C} \to \mathcal{C}\times \mathcal{C}\colon A \mapsto (A,A)\colon f \mapsto (f,f) \).

A natural transformation \(\eta\colon F\to G\) between functors \( F, G\colon \mathcal{C} \to \mathcal{D} \) is a family of morphisms \( \{ \eta_X\in \mathcal{D}\colon F(X) \to G(X) \}_{X\in \mathcal{C}} \) that makes the following diagram commutes for all objects \( X, Y\in \mathcal{C} \) and for any morphisms \( f \).

Each morphism \( \eta_X \) is called the component of \( \eta \) at \( X \) or the \( X \) component of \( \eta \).

• Natural Equivalence, Isomorphism of Functors

For every object \(X\) in \(\mathcal{C}\), the morphism \(\eta_X\) is an isomorphism in \(\mathcal{D}\).

Two natural transformations \( \eta\colon F\Rightarrow G \) and \( \epsilon\colon G\Rightarrow H \) between functors \( F, G, H\colon \mathcal{C} \to \mathcal{D} \) can be composed component-wise: \[ \epsilon \circ \eta\colon F \Rightarrow H, \quad (\epsilon\circ \eta)_X = \epsilon_X\circ \eta_X. \]

The name comes from the vertical arrangement of natural transformations in

where the composed natural transformation is "longer".

The two natural transformation \( \eta\colon F \Rightarrow G \) and \( \epsilon\colon J \Rightarrow K \) with \( F, G\colon \mathcal{C} \to \mathcal{D} \) and \( J, K\colon \mathcal{D} \to \mathcal{E} \), can be composed using the functor composition: \[ \epsilon * \eta\colon J\circ F \Rightarrow K \circ G, \quad (\epsilon*\eta)_X = \epsilon_{G(X)}\circ J(\eta_X) = K(\eta_X) \circ \epsilon_{F(X)}. \]

The composition makes the composed natural transformation "wider".

The horizontal composition is easily understood by a commutative diagram.

This is one step further from the commutative diagram in the definition of a natural transformation. Notice that the horizontal composition is nothing but the natural isomorphism from the \( J\circ F \) "column" to the \( K\circ G \) "column".

Whiskering can be used to simplify the notation. It is a binary operation between a functor and a natural transformation:

\begin{align*} J\eta\colon J\circ F \to J \circ G, \quad (J\eta)_X &:= J(\eta_X), \\ \eta J\colon J\circ F \to K \circ F, \quad (\eta J)_X &:= \eta_{J(X)}.\\ \end{align*}

In light of whiskering, we can write \( \epsilon * \eta = \epsilon G \circ J\eta = K\eta\circ \epsilon F \), where \( \circ \) is the vertical composition.

The vertical composition and horizontal composition interchange in the diagram:

\( (\beta' \circ \alpha') * (\beta \circ \alpha ) = (\beta' * \beta) \circ (\alpha' * \alpha) \).

It can be seen relatively easily in the commutitive diagram with the shape of 3×3×2 cuboid.

A coproduct of two objects \(X_{1}\) and \(X_{2}\) is an object \(X_{1}\oplus X_{2}\) (also written as \(X_1\sqcup X_2\), \(X_1\coprod X_2\) or \(X_1 + X_2\)), such that for every object \(Y\) and every morphisms \(f_1\) and \(f_2\), there exists a unique morphism \(f\) that makes the following diagram commutes:

Coequaliser is a colimit \((Q, q)\) of the diagram

• If \( \mathcal{J} \) is a discrete category, the limit is the product.
• If \( \mathcal{J} \) looks like \( \bullet \rightrightarrows \bullet \), the limit is the equaliser.
• If \( \mathcal{J} \) looks like \( \bullet \rightarrow \bullet \leftarrow \bullet \), the limit is the the pullback.

Colimit is the limit in the opposite category.

• The limit of a diagram \( F\colon \mathcal{J} \to \mathcal{C} \) is the universal morphism from the diagonal functor \( \Delta\colon \mathcal{C} \to \mathcal{C}^{\mathcal{J}} \) to \( F \).
  • The diagram is represented by the functor \( F \) and the limit cone (cone that is also the limit) is compressed into the natural transformation between the constant functor \( \Delta(\lim F) \) to \( F \).
• Similarly, the colimit of \( F\colon \mathcal{J} \to \mathcal{C} \) is a universal morphism from \( F \) to \( \Delta \).
• The tensor algebra \( T(V)\in K\text{-} \mathbf{Alg} \) of a vector space \( V \in K\text{-}\mathbf{Vect} \) over a field \( K \), with the inclusion map \( \iota\colon V \to U(T(V)) \) where \( U\colon K\text{-} \mathbf{Alg} \to K\text{-} \mathbf{Vect}\) is a forgetful functor, is the universal morphism from \( V \) to \( U \).

In a locally small category \( \mathcal{C} \), given a functor \( F: \mathcal{C} \to \mathbf{Set} \), and any object \( A \in \mathcal{C} \), there is one-to-one correspondence between natural transformations from the hom functor to \( F \), and the elements of \( F(A) \):

\begin{equation*} \mathrm{Hom}(\mathrm{Hom}(A,-), F) \cong F(A). \end{equation*}

Informally, every property of the object \( A \) is recoverable by considering many morphisms from objects \( X \) to \( A \), \( \{X\to A\} \).

A natural transformation \( \Phi \) is completely determined by choosing \( u \in F(A) \) that \( \mathrm{id}_A \in \mathrm{Hom}(A,A)\) maps to.

• \( \{\mathbf{1}\to A\} \) is isomorphic to \( \mathrm{ob}(A) \).
• \( \{(0,1)\to A\} \) is every open set in \( A \) .

A presheaf \( F \) is a contravariant functor from a posetal category \( \mathcal{C} \) to \( \mathbf{Set} \).

A sheaf \( F \) is a presheaf that for every open cover \( \{U_i\} \) of any open set \( U \) \( F(U) \) is an in the following diagram: \[ F(U) \overset{f}{\to} \prod_i F(U_i) \overunderset{g}{h}{\rightrightarrows} \prod_{i,j} F(U_i\cap U_j) \] where \( f \) is the product of the restriction maps \[ f := \prod_ir^U_{U_i}\colon \sigma \mapsto (\sigma|_{U_1}, \sigma|_{U_2}, \dots) \] and \( g, h \) are the product of further restrictions

\begin{align*} g &:= \prod_{i,j} r^{U_i}_{U_i\cap U_j}, \\ h &:= \prod_{i,j} r^{U_j}_{U_i\cap U_j}. \end{align*}

Intuitively, \( F(U) \) is all the continuous functions that is defined piecewise on each \( U_i \), provided that the pieces match at their intersections \( U_i\cap U_j \).

The natural functor from the category of presheaf \( \mathbf{PSh}(\mathcal{C}) \) to the category of sheaf \( \mathbf{Sh}_J(\mathcal{C}) \). What is considered "natural" is dependent on the structure called coverage \( J \) of the posetal category \( \mathcal{C} \).

A monad is a triple \( (T, \eta, \mu) \) consisting of a endofunctor \( T\colon \mathcal{C} \to \mathcal{C} \) and two natural transformations \( \eta\colon 1_{\mathcal{C}} \to T, \mu\colon T^2 \to T \), where \( 1_{\mathcal{C}} \) is the identity functor, that satisfies the following commutative diagrams:

The first diagram is asserting the associativity of binary operation \( \mu \), and the second diagram is asserting the existence of an identity element \( \eta(1_{\mathcal{C}}) \) of the operation \( \mu \). Thus "monoid in the (monoidal) category of endofunctors (under composition)", with the understanding of monoid in the category theoretic sense.

• In functional programming, the Maybe monad is an endofunctor on the category of types.

Given a monad \( (T, \eta, \mu) \) on a category \( \mathcal{C} \), the Kleisli category \( \mathcal{C}_T \) of \( \mathcal{C} \) can be constructed as follows:

\begin{align*} & \mathrm{ob}(\mathcal{C}_T) := \mathrm{ob}(\mathcal{C}), \\ & \mathrm{Hom}_{\mathcal{C}_T}(X,Y) := \mathrm{Hom}_{\mathcal{C}}(X, TY) \end{align*}

and the composition of morphisms \( f\colon X\to TY \) and \( g\colon Y\to TZ \) is given by \[ g\circ_T f := \mu_Z \circ Tg \circ f\colon X \to TY \to T^2Z \to TZ \] with the identity morphism being \( \eta_X \).

Given a monad \( (T, \eta, \mu) \) on a category \( \mathcal{C} \), the Eilenberg-Moore category \( \mathcal{C}^T \) can be constructed. The objects of \( \mathcal{C}^T \) are the \( T \)-algebras \( (x,h) \) where \( x \) is an object of \( \mathcal{C} \) and structure map of the algebra \( h\colon Tx \to x \) that makes the following diagrams commute:

The morphism \( f\colon (x,h) \to (x', h') \) of \( T \)-algebras is the morphism \( f\colon x\to x' \) such that the following diagram commutes:

A monoid \( (M, \eta, \mu) \) in a monoidal category \( (\mathcal{C}, I, \otimes) \) is an object \( M\in \mathcal{C} \) equipped with two morphisms \( \eta, \mu \) that satisfies the pentagon diagram and unitor diagram.