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Epistemology - m a t h 4 w i s d o m - g m a i l
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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
____ - 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.
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: {$f = G(\bar{f})\circ\eta_D$}
{$C=F(D)$} 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 {$\eta_D=G(\bar{\eta}_D)\circ\eta_D$}. |

This page was last changed on January 13, 2022, at 08:46 PM