Intro to category theory [categories]

    The world of flora

    For the purpose of this project, we want to be able to talk about things in several different contexts, so I will introduce a theory of categories here. First, we take our logic to be a classical first-order logic with equality. Here I will present the first

    Definition. Objects, morphisms, and categories [#]

      categories, objects and morphisms are primitive notions, and we say is an object of and write in to to say that is a morphism from a unique source to a unique target in . If we have already chosen some category, or the math we are doing is for an arbitrary category, we may choose to omit “in “

      Axiom. Identity morphisms [identity-morphism]

        For every object of a category , there is a morphism in called the identity morphism on .

        Axiom. Composition [composition]

          For each composable pair of morphisms in has exactly one morphism called their composition.

          Axiom. Associativity of Composition [composition-associativity]

            For composable morphisms , .

            Axiom. Identity laws [identity-laws]

              For every , .

              Now with these first four axioms, we can start introducing some definitions and doing some proofs. First we will show that the identity morphism is in fact unique which justifies the notation.

              Lemma. The identity morphism is unique [uniqueness-identity]

                Proof. [#]

                  Let be arbitrary. Suppose and are identity morphisms. Then as desired.

                  We then define inverses, and isomorphism.

                  Definition. Inverses and isomorphism [inverses-and-isomorphism]

                    If , are morphisms.

                    • is a left inverse of when .
                    • is a right inverse of when .
                    • is a section if it has a left inverse.
                    • is a retraction if it has a right inverse.
                    • is an isomorphism if it is both a section and a retraction. We will also introduce the notation for an isomorphism .

                    Lemma. The inverse of an isomorphism is unique. [uniqueness-inverse]

                      Let be an isomorphism. Let be its left inverse and be its right inverse. Then .

                      Proof. [#]

                        We have by definition that and . Then we have as desired.

                        Now, with the inverse of an isomorphism being unique we can name the inverse of an isomorphism.

                        Definition. Inverse of an isomorphism [inverse-isomorphism]

                          If is an isomorphism we define to be the inverse of . This notation is justified above

                          We then define what it means for objects to be isomorphic.

                          Definition. Isomorphic objecs [isomorphic-objects]

                            We say that objects are isomorphic, written if there exists an isomorphism .

                            We now prove a couple of results that will be useful to show that is an equivalence relation.

                            Lemma. The composite of isomorphisms is an isomorphism [isomorphism-composite]

                              If we have and then .

                              Proof. [#]

                                Consider . Then and as desired.

                                Thus .

                                Lemma. The identity is an insomorphism [isomorphism-identity]

                                  For any we have .

                                  Proof. [#]

                                    Let be an arbitrary object; then there is an identity . We then have that , and thus it is an isomorphism as desired.

                                    With these lemmas out of the way we can now show that is an equivalence relation.

                                    Theorem. is an equivalence relation [isomorphic-equivalence]

                                      is an equivalence relation: it is

                                      • Reflexive:
                                      • Symmetric:
                                      • Transitive:

                                      Proof. [#]

                                        (Reflexivity): Consider an object . Then by .

                                        (Symmetry): Suppose for some objects . Then there is an isomorphism . Thus by .

                                        (Transitivity): Suppose . Then there are . Thus by .

                                        We now introduce monomorphisms and epimorphisms.

                                        Definition. Monomorphisms and epimorphisms [monos-and-epis]

                                          Given :

                                          • is a monomorphism (or monic) when for any ,

                                          • is an epimorphism (or epic) when for any ,

                                          Lemma. Sections are monic and retractions are epic [monic-sections-epic-retractions]

                                            If is a section and is a retraction then is monic and is epic.

                                            Proof. [#]

                                              (Sections): Since is a section we can find such that . Now given arbitraty with we have that . Then and thus as desired.

                                              (Retractions): This proof is left as an exercise to the reader.

                                              Corollary. Isomorphisms are both monic and epic [monic-epic-isomorphisms]

                                                If then is monic and epic.