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Introduction E9F5FC
Questions FFFFC0
Software
Classify adjunctions, Examples of adjunction, Induction restriction adjunction, Adjunction in statistics, Category theory, Limits vs colimits, Equivalence, Sameness, Definition of adjunction, Nonexistence of adjunction
Given a functor, construct its left adjoint and right adjoint, or show that they do not exist.
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- Base the construction on the definition of adjunction in terms of the universal mapping property.
- Consider what we can say about the construction in simple cases where the category has a single object, has at most one morphism per pair of objects, or has no morphism.
Note that given a functor, its right adjoints are isomorphic by virtue of the Yoneda lemma. (Adowey, 9.9) Likewise, the left adjoint is unique up to isomorphism.
Definition in terms of universal mapping property
Given functor {$G:\mathcal{C}\rightarrow \mathcal{D}$}. Its left adjoint {$F$} needs to satisfy the following.
- For every {$D\in\mathcal{D}$}
- Define {$F(D)$} in {$\mathcal{C}$} such that:
- There is a well defined morphism {$\eta_D:D\rightarrow G(F(D))$} such that:
- For every morphism {$f:D\rightarrow G(C)$}
- There exists a unique morphism {$\bar{f}:F(D)\rightarrow C$}
- By which it factors: {$f = G(\bar{f})\circ\eta_D$}
Case where {$\mathcal{C}$} has a single object {$C$} and {$\mathcal{D}$} has a single object {$D$}.
{$C=F(D)$} and {$D=G(F(D))=G(C)$}
In {$\mathcal{D}$}, all of the morphisms start and end at {$D$}, and among them we have {$f=G(\bar{f})\circ\eta_D$}.
In particular, we have {$\eta_D=G(\bar{\eta}_D)\circ\eta_D$}.