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I need to determine if the forgetful functor \begin{equation} U: Set_\star \longrightarrow Set \end{equation} that forgets "the base points" has left adjoint or right adjoint, but I'm struggling to even defining a functor $Set \longrightarrow Set_\star$ because I don't know how to choose the base point (If I choose for example as basepoint any element on the set, the functor would not be well-defined).

Can anyone help me? Thanks

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Hint : have you thought about adding a basepoint, rather than choosing one ?

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  • $\begingroup$ I would still need to choose how to add this basepoint though... For a morphism $f: X \longrightarrow Y$ I added as a basepoint a point $d$ and defined $f': X\cup \{d\} \longrightarrow Y \cup \{d\}$ using $f$ and $f'(d)=d$, but then I might have trouble with compositions. I would need like a "universal" basepoint, wouldn't I? $\endgroup$ – square17 Oct 12 '20 at 7:16
  • $\begingroup$ Why might you have trouble with composition ? $\endgroup$ – Maxime Ramzi Oct 12 '20 at 7:17
  • $\begingroup$ Well, I cannot add just any point that is not in $X$ not $Y$, for example, if I try to do the composition with $g : Y \longrightarrow Z$ in set, it would be fine, but then $g': Y \cup \{d'\} \longrightarrow Z \cup \{d'\}$ might not be possible if $d \neq d'$, so I might not be able to do the composition $g' \circ f'$ and it would not be a functor $\endgroup$ – square17 Oct 12 '20 at 7:24
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    $\begingroup$ Oh I think you misunderstood: you're supposed to add different basepoints to different sets. For each set $X$, add a basepoint $*_X$ which is not in $X$. Then define $X_+ = X\cup \{*_X\}$ and $f_+ :X_+\to Y_+$ by $f_+(x) = f(x), f_+(*_X) = *_Y$. Then there's no composition problem $\endgroup$ – Maxime Ramzi Oct 12 '20 at 7:36
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    $\begingroup$ I don't think it's a right adjoint. For it to be a right adjoint to $U$, you would need a co-unit transformation $U(X_+)\to X$, which would amount to a choice of a basepoint in $X$ for every $X$ ! $\endgroup$ – Maxime Ramzi Oct 12 '20 at 9:28

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