Index. Category Theory [005J]
Index. Category Theory [005J]
This index collects some of my notes on the book Category Theory in Context by Emily Riehl.
Definition 1. Category [005K]
Definition 1. Category [005K]
A category is a collection of objects with a collection of morphisms with the following properties
- Each morphism has a domain and codomain . We write .
- Each object has an identity morphism .
- For any morphisms there exists a composite morphism
- For any , both and are equal to .
- For any composable triple of morphisms , , , the composition is associative, e.g . We will denote it by
We will often call morphisms arrows.
Definition 4. Monoid [0062]
Definition 4. Monoid [0062]
A monoid is a one object category.
Definition 7. Isomorphism in a Category [005W]
Definition 7. Isomorphism in a Category [005W]
An isomorphism is a morphism for which there exists a morphism such that and . In that case we say that and are isomorophic , in symbols .
Lemma 10. Inverse isomorphisms are unique [006B]
Lemma 10. Inverse isomorphisms are unique [006B]
the left inverse and right inverse to a common morphism coincide. This means every morphism can have at most one inverse isomorphism.
Proof. TODO
Definition 12. Automorphism [0069]
Definition 12. Automorphism [0069]
An endomorphism that is also an isomorphism is called an an automorphism.
Definition 13. Groupoid [0065]
Definition 13. Groupoid [0065]
A groupoid is a category in which every morphism is an isomorphism.
Lemma 16. Maximal groupoids are subcategories [006D]
Lemma 16. Maximal groupoids are subcategories [006D]
Let be a category. The maximal groupoid inside defines a subcategory.
Proof. TODO
Definition 17. Opposite Category [006E]
Definition 17. Opposite Category [006E]
For a category , the opposite category is given by the same objects and morhpisms as in , where every morphism in is defined to have opposite domain and codomain compared to the corresponding morphism in . For each object , serves as the identity in
We define composition to be . Observe that and are precisely composable when and are composable.
Remark 18. opposite categories are categories [006F]
Remark 18. opposite categories are categories [006F]
Note that for every category , its opposite category fullfills all axioms of a category.
Proof. TODO
TODO
- retraction (left inverse)
- section ( right inverse)
- automorphism
- groupoid
- asubcategory
- maximal subgroupoid