right adjoint functor and objectwise representable

1. Proposition

Let \(\mathcal{C}, \mathcal{D}\) be categories, \(\mathcal{R}: \mathcal{C} \rightarrow \mathcal{R}\) a functor. TFAE:

  1. \(\mathcal{R}: \mathcal{D} \rightarrow \mathcal{C}\) is a right adjoint to the left adjoint \(\mathcal{L}\)
  2. for each \(d \in \mathrm{Ob}(\mathcal{D})\), the functor
\begin{align*} \mathrm{Hom}_{\mathcal{D}}(d, R(-)) \end{align*}

is representable functor

2. Proof

2.1. 1) \(\implies\) 2)

2.2. 2) \(\implies\) 1)

we construct a functor using

\begin{align*} X \mapsto& \mathrm{Repr}(X) \end{align*}

take isomorphism, by yoneda given by element

Date: nil

Author: Anton Zakrewski

Created: 2024-10-15 Di 23:17