2022-06-12

300: Covariant Functor

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A definition of covariant functor

Topics


About: category
About: functor

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of covariant 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) \(\mathcal{F} (f_1) \in Mor (\mathcal{F} (O_1), \mathcal{F} (O_2))\); 2) \(\mathcal{F} (\mathbb{1}_{O_1}) = \mathbb{1}_{\mathcal{F} (O_1)}\); 3) \(\mathcal{F} (f_2 \circ f_1) = \mathcal{F} (f_2) \circ \mathcal{F} (f_1)\)


2: Note


There is also 'contravariant functor'.


References


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