Category Theory

Table of Contents

1. Category

1.1. Definition

  • A category \(\mathcal C\) consists of objects \(\mathrm{ob}(\mathcal C)\), morphisms \(\mathrm{hom}_{\mathcal{C}}(A,B)\) for all pairs of objects \(A,B\) in \(\mathcal{C}\), and binary operation \(\circ\) called composition of morphisms
  • which satisfies
    • 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 \(h\circ(g\circ f) = (h\circ g)\circ f.\)

1.2. Classification

1.2.1. Small Category

1.2.2. Locally Small Category

  • The homset is not a proper class.

1.3. Functor Category

  • \(\mathcal{D}^\mathcal{C}\)
  • The class of functors from \(\mathcal{C}\) to \(\mathcal{D}\) with morphisms being the natural transformations.

2. Morphism

2.1. Classification

2.1.1. Monomorphism

  • \(\hookrightarrow\) (Inclusion), \(\rightarrowtail\) (Injection)
  • Monic Morphism, Mono
  • Left-Cancellative: \[ f\circ g_1 = f\circ g_2 \implies g_1 = g_2 \]
2.1.1.1. Regular Monomorphism
  • Equalizer of some parallel morphisms
  • The limit \((E, eq)\) of the morphisms \((O, m)\) that equalizes \(f, g\colon X\to Y\)
  • equalizer.png
  • This is the universal property of the equalizer of \(f, g\): \(\mathrm{Eq}(f,g) := \{x\in X \mid f(x) = g(x)\}\)
  • Equaliser (mathematics) - Wikipedia
2.1.1.2. Extremal Monomorphism
  • \(\mu = \varphi \circ \varepsilon\land \varepsilon\text{ epimorphism} \implies \varepsilon\text{ isomorphism}\)
2.1.1.3. Immediate Monomorphism
  • \(\mu = \mu' \circ \varepsilon\land \mu'\text{ monomorphism} \land \varepsilon\text{ epimorphism} \implies \varepsilon\text{ isomorphism}\)
2.1.1.4. Strong Monomorphism
  • \(\mu\) is strong if the following diagram commutes

strongmono.png

Figure 1: strongmono

2.1.1.5. Split Momomorphism
  • \(\exists \varepsilon : \varepsilon \circ \mu = 1\)

2.1.2. Epimorphism

  • \(\twoheadrightarrow\) (Surjection, Projection)
  • Epic Morphism
  • Right-Cancellative: \[ g_1\circ f = g_2\circ f \implies g_1 = g_2. \]
2.1.2.1. Regular Epimorphism

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

    coequalizer.png

    Figure 2: coequal

2.1.2.2. Extremal Epimorphism
  • \(\varepsilon = \mu \circ \varphi \land \mu\text{ monomorphism} \implies \mu\text{ isomorphism}\)
2.1.2.3. Immediate Epimorphism
  • \(\varepsilon = \mu \circ \varepsilon'\land \mu\text{ monomorphism}\land \varepsilon'\text{ epimorphism} \implies \varepsilon\text{ isomorphism}\)
2.1.2.4. Strong Epimorphism
  • \(\varepsilon\) is strong if the following diagram commutes

strongepic.png

Figure 3: strongepic

2.1.2.5. Split Epimorphism
  • \(\exists \mu : \varepsilon \circ \mu = 1\)

2.1.3. Bimorphism

  • \(\mathrel{\rightarrowtail\kern-10pt\twoheadrightarrow}\)
  • Both monomorphism and epimorphism

2.1.4. Isomorphism

  • \(\xrightarrow{\mathpunct{\sim}}\)
  • \(\exists g : g\circ f = 1_X \land f\circ g = 1_Y\)
  • See isomorphism.

2.1.5. Endomorphism

  • \(\mathrm{end}(X)\)
  • \(f\colon X\to X\)

2.1.6. Automorphism

  • Both endomorphism and isomorphism

2.1.7. Retraction

  • Right inverse exists
  • \(\exists g : f\circ g = 1_Y\)

2.1.8. Section

  • Left inverse exists
  • \(\exists g : g\circ f = 1_X\)

2.2. Properties

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

2.3. Kernel

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

2.4. Cokernel

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

2.5. Image

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

2.6. Coimage

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

3. Functor

  • \(F\colon \mathcal{C}\to \mathcal{D}\)
  • It maps objects and morphisms, such that:
    • Covariant: \(F(f)\colon F(X) \to F(Y)\) / Contravariant: \(F(f)\colon F(Y) \to F(X)\)
    • \(F(1_X) = 1_{F(X)}\)
    • \(F(g\circ f) = F(g)\circ F(f)\)

4. Commutative Diagram

  • It is a diagram in which commuting different composition of morphisms is allowed.

5. Natural Transformation

  • \(\eta\colon F\to G\)
  • Association between every object \(X\) in \(\mathcal{C}\) and morphism \(\eta_X\colon F(X) \to G(X)\), such that the following diagram commutes

naturaltransform.png

5.1. Natural Isomorphism

  • Natural Equivalence, Isomorphism of Functors
  • For every object \(X\) in \(\mathcal{C}\), the morphism \(\eta_X\) is an isomorphism in \(\mathcal{D}\).

6. Universal Property

  • It is a property that is satisfied over all contructions.
  • The construction that satisfies the universal property is called the universal construction.
  • The relational property that uniquely determines an object up to isomorphism, since the category theory does not deal with the internal structure of an object.
  • A universal morphism from \(X\) to \(F\) is a unique pair \((A, u\colon X\to F(A))\) in \(\mathcal{D}\), which has the following property called universal property
  • universalproperty.png
  • By reversing the arrows, one get the dual of the property.

7. Product

  • \[ X_1 \times X_2 \]
  • A product of two objects \(X_{1}\) and \(X_{2}\) is an object \(X_{1}\times X_{2}\), such that for every object \(Y\) and every morphisms \(f_1\) and \(f_2\), there exists a morphism \(f\) that satisfies the following commutative diagram: CategoricalProduct-03.svg
  • "The smallest object that contains both object"

7.1. Coproduct

  • 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: Coproduct-03.svg

8. Diagram

  • Diagram of type \(\mathcal{J}\) in a category \(\mathcal{C}\) is a covariant functor \(F\colon \mathcal{J}\to \mathcal{C}.\)
  • \(\mathcal{J}\) is called the index category, the scheme or the shape.

9. Limit

  • The limit of the diagram \(F\colon \mathcal{J}\to \mathcal{C}\) is a cone \((L, \{\phi_X\})\) to \(F\) such that factors every cone \((N, \{\psi_X\})\) by a unique morphism called the mediating morphism \(u\colon N\to L\)
  • Functor_cone_(extended).svg
  • This can be understood as the aptest cone that is universal.

9.1. Colimit

Colimit is the limit in the opposite category.

10. Yoneda Lemma

10.1. Statement

For a locally small category \( \mathcal{C} \), a functor \( F: \mathcal{C} \to \mathbf{Set} \), and any object \( A \in \mathcal{C} \), there's one-to-one correspondence between natural transformations and elements of \( A \):

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

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

10.2. Examples

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

11. See Also

12. References

Created: 2025-05-06 Tue 23:34