Definition:Category of Relations
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Definition
The category of (binary) relations, denoted $\mathbf{Rel}$, is the metacategory with:
| Objects: | sets | |
| Morphisms: | binary relations $\RR \subseteq A \times B$. | |
| Composition: | composition of relations | |
| Identity morphisms: | identity mappings |
Note
The reason to call $\mathbf{Rel}$ a metacategory is foundational; allowing it to be a category would bring us to axiomatic troubles.
Also see
- Category of Relations is Category
- Definition:Category of Sets
- Results about the category of relations can be found here.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.4$