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Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor

$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$

is corepresentable by some object $L(X)$.

Since adjoints in category theory are enormously widespread, I would like to ask: what are useful / noteworthy examples, mainly from the perspective of teaching, of ''partial'' adjoints?

Examples I can find so far are:

  • limits being a partial right adjoint of the diagonal functor, which may not be a right adjoint unless all limits exist,
  • the discrete-topology functor $Sets \to Top$ admits a partial left adjoint given by connected components $\pi_0$, defined on locally connected spaces.
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  • $\begingroup$ Similarly, right kan extensions to $C$ along a functor $f:A \to B$ are a partial right adjoint to the restriction functor $[B,C] \to [A,C]$, which may not be an adjoint unless all right Kan extensions exist. $\endgroup$
    – john
    Commented Nov 21, 2021 at 17:59

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