definition of congruence on 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
Starting Context
- The reader knows a definition of category.
Target Context
- The reader will have a definition of congruence on 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:
\( C\): \(\in \{\text{ the categories }\}\)
\(*\sim\): \(\in \{\text{ the equivalence relations on } Mor (C)\}\)
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Conditions:
\(f_1 \sim f_2 \implies \exists O_1, O_2 \in Obj (C) (f_1, f_2 \in Mor (O_1, O_2))\)
\(\land\)
\(\forall O_1, O_2, O_3 \in Obj (C) (\forall f_1, f_2 \in Mor (O_1, O_2), \forall f_3, f_4 \in Mor (O_2, O_3) ((f_1 \sim f_2 \land f_3 \sim f_4) \implies (f_3 \circ f_1 \sim f_4 \circ f_2)))\)
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2: Natural Language Description
For any category, \(C\), any equivalence relation on \(Mor (C)\), \(\sim\), such that \(f_1 \sim f_2 \implies \exists O_1, O_2 \in Obj (C) (f_1, f_2 \in Mor (O_1, O_2))\) and \(\forall O_1, O_2, O_3 \in Obj (C) (\forall f_1, f_2 \in Mor (O_1, O_2), \forall f_3, f_4 \in Mor (O_2, O_3) ((f_1 \sim f_2 \land f_3 \sim f_4) \implies (f_3 \circ f_1 \sim f_4 \circ f_2)))\)
