2022-06-05

81: Category

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definition of category

Topics


About: category

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of category.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description


Here is the rules of Structured Description.

Entities:
∗C: = any collection of objects ,Obj(C)={Oα|α∈A}, where A is a possibly uncountable index set, with any collection of morphisms, Mor(C)={Mor(O1,O2)|O1,O2∈Obj(C)}
Conditions:
∀O1,O2,O3,O4∈Obj(C),∀f1∈Mor(O1,O2),∀f2∈Mor(O2,O3),∀f3∈Mor(O3,O4)
(
1) f2∘f1∈Mor(O1,O3)
2) ∃idOj∈Mor(Oj,Oj)(f1∘idO1=f1∧idO2∘f1=f1)
3) f3∘(f2∘f1)=(f3∘f2)∘f1
)
//

2: Natural Language Description


Any collection of objects, Obj(C)={Oα|α∈A}, where A is a possibly uncountable index set, with any collection of morphisms, Mor(C)={Mor(O1,O2)|O1,O2∈Obj(C)}, such that for each O1,O2,O3,O4∈Obj(C) and for each f1∈Mor(O1,O2),f2∈Mor(O2,O3),f3∈Mor(O3,O4), 1) f2∘f1∈Mor(O1,O3); 2) ∃idOj∈Mor(Oj,Oj)(f1∘idO1=f1∧idO2∘f1=f1); 3) f3∘(f2∘f1)=(f3∘f2)∘f1

3: Note


Typically, each object is a set and each morphism is a map, but they are generalized to be not necessarily so.

References
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