A definition of category
Topics
About: category
The table of contents of this article
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: Definition
Any collection of objects, \({O_\alpha}\), and any collection, \(Mor (O_1, O_2)\), of morphisms for each ordered pair of objects, \((O_1, O_2)\), such that for any \(f_1 \in Mor (O_1, O_2)\), \(f_2 \in Mor (O_2, O_3)\), and \(f_3 \in Mor (O_3, O_4)\), 1) \(f_2 \circ f_1 \in Mor (O_1, O_3)\); 2) for each object, \(O\), there is an identity morphism, \(\mathbb{1}_O \in Mor (O, O)\), which means that \(f_1 \circ \mathbb{1}_{O_1} = f_1\) and \(\mathbb{1}_{O_3} \circ f_2 = f_2\); 3) \(f_3 \circ (f_2 \circ f_1) = (f_3 \circ f_2) \circ f_1\)
2: Note
Typically, an object is a set, and a morphism is a map, but they are generalized to be not necessarily so.
