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

2008-12-09T01:29:18Z

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

New page

==Definition==

Suppose <math>\mathcal{C}</math> and <math>\mathcal{D}</math> are [[category|categories]]. Suppose <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> and <math>\mathcal{G}:\mathcal{D} \to \mathcal{C}</math> are [[functor]]s. We say that <math>\mathcal{F}</math> is a '''left-adjoint functor''' to <math>\mathcal{G}</math> (equivalently, <math>\mathcal{G}</math> is a '''right-adjoint functor''' to <math>\mathcal{F}</math>) if for every <math>A \in \operatorname{Ob}\mathcal{C}, B \in \operatorname{Ob}\mathcal{D}</math>, we can define a map:

<math>\eta_{AB}: \mathcal{D}(\mathcal{F}A,B) \to \mathcal{C}(A,\mathcal{G}B)</math>

such that <math>\eta_{AB}</math> is a [[natural transformation]] in each variable, keeping the other fixed. More precisely:

{{fillin}}

Adjoint functors - Revision history

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

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