A definition of contravariant functor
Topics
About: category
About: functor
The table of contents of this article
Starting Context
- The reader knows a definition of category.
Target Context
- The reader will have a definition of contravariant functor.
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
For any categories, \(\mathcal{C}_1\) and \(\mathcal{C}_2\), any map, \(\mathcal{F}: \mathcal{C}_1 \rightarrow \mathcal{C}_2\), such that for any objects, \(O_1, O_2, O_3 \in \mathcal{C}_1\), and any morphisms, \(f_1 \in Mor (O_1, O_2)\) and \(f_2 \in Mor (O_2, O_3)\), 1) \(\mathcal{F} (f_1) \in Mor (\mathcal{F} (O_2), \mathcal{F} (O_1))\); 2) \(\mathcal{F} (\mathbb{1}_{O_1}) = \mathbb{1}_{\mathcal{F} (O_1)}\); 3) \(\mathcal{F} (f_2 \circ f_1) = \mathcal{F} (f_1) \circ \mathcal{F} (f_2)\)
2: Note
There is also 'covariant functor'.