Vipul: New page: {{basicdef}} ==Definition== Suppose $\mathcal{C},\mathcal{D}$ are categories, and $\mathcal{F}:\mathcal{C} \to \mathcal{D}$ and $\mathcal{G}:\math...$

2008-12-09T00:58:53Z

New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C},\mathcal{D}</math> are categories, and <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> and <math>\mathcal{G}:\math...

New page

{{basicdef}}

==Definition==

Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]], and <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> and <math>\mathcal{G}:\mathcal{C} \to \mathcal{D}</math> are [[functor]]s. A '''natural transformation''' <math>\eta:\mathcal{F} \to \mathcal{G}</math> is defined by the following data:

For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a morphism <math>\eta_A \in \mathcal{D}(\mathcal{F}A, \mathcal{G}A)</math>,

satisfying the following compatibility condition: For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, and any <math>f \in \mathcal{C}(A,B)</math>, we have:

<math>\mathcal{G}f \circ \eta_A = \eta_B \circ \mathcal{F}f</math>.

Natural transformation - Revision history

Vipul: New page: {{basicdef}} ==Definition== Suppose \mathcal{C},\mathcal{D} are categories, and \mathcal{F}:\mathcal{C} \to \mathcal{D} and \mathcal{G}:\math...

Vipul: New page: {{basicdef}} ==Definition== Suppose $\mathcal{C},\mathcal{D}$ are categories, and $\mathcal{F}:\mathcal{C} \to \mathcal{D}$ and $\mathcal{G}:\math...$