Extracurriculars
Notation:
Morphism
Parallel pair of morphisms
Opposing pair of morphisms
“maps to”
“yields” / “leads to”
Implied existece of another morphism
Genuine equality
Definitional equality
Isomorphism
Weaker equivalence
Epimorphism
Inclusion/Embedding/“beinhaltet”/ ?
| Term | Definition |
|---|---|
| Commutative | The order of the elements in an operation does not affect the result |
| Commutative diagram | Diagram where all directed paths with the same starting and ending points yield the same result |
| Natural | |
| Functor |
Groups
| Term | Definition |
|---|---|
| Group | A non-empty set with a binary operation satisfying associativity, identity and inverse A groupoid with one object |
| Subgroup | |
| Group extension | |
| Abelian group | A Group equipped with an operation that satisfies the commutative property |
| Abelian extension | |
| Ring | |
| Quiver | A directed graph defined by objects and morphisms in a category |
Morphisms
| Term | Definition |
|---|---|
| Morphism | A structure-preserving map between two objects within a category |
| Homomorphism | A morphism mapping between two algebraic structures while preserving their operations |
| Isomorphism |
A morphism for which there exists a morphism so that A morphism which is both a monomorphism and an epimorphism. * |
| Endomorphism | A morphism whose domain equals its codomain |
| Automorphism | An isomorphic endomorphism |
Categories
| Term | Definition |
|---|---|
| Category |
Consists of objects and morphisms , such that:
|
| Subcategory | Subcollection of objects and morphisms of a category, such that it contains the domain and codomain of any morphism, the identity morphism of any object and the composite of any composable pair of morphisms in the subcategory |
| Small category | A category is small if it has only a set’s worth of arrows |
| Locally small category | A category is locally small if between any pair of objects there is only a set’s worth of morphisms |
| Hom-set | Set of arrows between a pair of fixed objects in a locally small category |
| Discrete Category | Every morphism is an identity |
| Slice category |
Groupoids
| Term | Definition |
|---|---|
| Groupoid | A category in which every morphism is an isomorphism has itself as its opposite category. Example: Discrete Category |
| Fundamental groupoid | For any space , its fundamental groupoid is a category whose objects are the points of and whose morphisms are endpoint-preserving homotopy classes of paths |
| Maximal groupoid | The subcategory containing all of the objects and only those morphisms that are isomorphisms |
| Comonoid |
| Not commutative | Commutative |
There should exist a Category , such that:
- Elements in : The same as in
- Morphisms in : The isomorphisms in
All identity morphisms can be taken into
Given isomorphisms , show that is isomorph.
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| Term | Definition |
|---|---|
| Duality | |
| Opposite Category | An opposite category of , called , has the same objects as and a morphism for each morphism in so that the domain of is defined to be the codomain of and the codomain of is defined to be the domain of |
| Composable | |
| Postcomposition | |
| Precomposition | |
| Monomorphism | |
| Epimorphism | |
| Supremum | |
| Infimum | |
| section/right inverse | |
| retraction/left inverse | |
| retract | |
| split morphism |
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| is an epimorphism if |
| is a monomoprhism if |
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NOTE: that doesn’t mean that . If so, then they would be isomorph. |
1.2.i.
Goals:
1.2.ii
1.2.iii
Fields are sets with Operations
https://abuseofnotation.github.io/category-theory-illustrated/02_category/
1.2.vi
Dual:
useful links & cool projects
OSDev wiki , osdever.net , it's good to be king , 9front (Plan 9) , mirageOS , Uxn , seL4 ,