definition of category
Topics
About: category
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of morphism.
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:
\(*C\): \(= \text{ any collection of objects }, Obj (C) = \{O_\alpha \vert \alpha \in A\}\), where \(A\) is a possibly uncountable index set, with any collection of morphisms, \(Mor (C) = \{Mor (O_1, O_2) \vert O_1, O_2 \in Obj (C)\}\)
Conditions:
\(\forall O_1, O_2, O_3, O_4 \in Obj (C), \forall f_1 \in Mor (O_1, O_2), \forall f_2 \in Mor (O_2, O_3), \forall f_3 \in Mor (O_3, O_4)\)
(
1) \(f_2 \circ f_1 \in Mor (O_1, O_3)\)
2) \(\exists id_{O_j} \in Mor (O_j, O_j) (f_1 \circ id_{O_1} = f_1 \land id_{O_2} \circ f_1 = f_1)\)
3) \(f_3 \circ (f_2 \circ f_1) = (f_3 \circ f_2) \circ f_1\)
)
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2: Natural Language Description
Any collection of objects, \(Obj (C) = \{O_\alpha \vert \alpha \in A\}\), where \(A\) is a possibly uncountable index set, with any collection of morphisms, \(Mor (C) = \{Mor (O_1, O_2) \vert O_1, O_2 \in Obj (C)\}\), such that for each \(O_1, O_2, O_3, O_4 \in Obj (C)\) and for each \(f_1 \in Mor (O_1, O_2), f_2 \in Mor (O_2, O_3), f_3 \in Mor (O_3, O_4)\), 1) \(f_2 \circ f_1 \in Mor (O_1, O_3)\); 2) \(\exists id_{O_j} \in Mor (O_j, O_j) (f_1 \circ id_{O_1} = f_1 \land id_{O_2} \circ f_1 = f_1)\); 3) \(f_3 \circ (f_2 \circ f_1) = (f_3 \circ f_2) \circ f_1\)
3: Note
Typically, each object is a set and each morphism is a map, but they are generalized to be not necessarily so.
