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What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying Mathematics. For the sake of clarity I would like you to follow a general scheme, a very naive example of which can be the following:

  1. Functors F and G between cats C and D
  2. Is the adjunction a (co)reflection?
  3. Does the left adjoint admit a left adjoint on its own?
  4. Anything you want to add

Obviously you are totally free to expand it, revert it...

I would also like to grasp something more than a mere enumeration: i.e. listing all adjunctions $\mathbf{Groups}\leftrightarrows\mathbf{Sets}$, $\mathbf{Monoids}\leftrightarrows\mathbf{Sets}$, $\mathbf{Mod}_R\leftrightarrows\mathbf{Sets}$ is certainly a good thing, but it would be slightly better to say that all these pairs come from a "general scheme of adjunction" $$ \text{generated object} \dashv \text{forgetful functor} $$ which can be (if I'm not wrong) studied for a general type of algebraic structure. Hence it would be better to write some sort of "reference card" about:

  1. The diagonal functor $\Delta_\mathbf J\colon \mathbf C\to \mathbf C^\mathbf J$ sending $C\in\text{Ob}_\mathbf C$ into the constant diagram over $C$ admits both a left and right adjoint (direct and inverse limit).
  2. Once you fixed a set $J$, here is an adjunction between $\mathbf{Sets}/J$ and $\mathbf{Sets}^J$ defined by functors $L\colon h\in \mathbf{Sets}/J\mapsto \big(h^\leftarrow(\{j\}\big)_{j\in J}$ and $M\colon \{H_j\}_{j\in J}\mapsto \big(\coprod_{j\in J} H_j\to J\big)\in \mathbf{Sets}/J$, which turns out to be an equivalence
  3. There exists an adjunction between $\mathrm{PSh}(X)$ and $\mathbf{Top}/X$ for any topological space $X$ ($\text{bundle of germs}\dashv\text{(pre)sheaf of sections}$), which turns out to be an equivalence if we restrict...
  4. Given a ring $R$ the functor $R[\;\;]\colon \mathbf{Groups}\to \mathbf{Rings}$ sending a group in its group ring admits a right adjoint, namely $U\colon R\mapsto R^\times$ (units in $R$).
  5. The inclusion functor $\mathbf{Kelley}\to\mathbf{Top}$ admits a right adjoint, the kelleyfication of a topological space
  6. (Following Gabriel&Zisman) The inclusion functor between (small) categories $\mathbf{cat}$ and (small) groupoids $\mathbf{Gpds}$, admits both a left adjoint ($\mathbf{C}\mapsto \mathbf{C}[\text{Mor}_\mathbf{C}^{-1}]$ in the notation used for the calculus of fractions) and a right adjoint ($\mathbf{C}\mapsto \mathbf{C}^\times$, sending a category in the groupoid obtained deleting every noninvertible arrow).
  7. ...

Feel free to say this is a silly or boring question.

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    Adjoint of Sets->Commutative Rings is free=polynomial ring.2011-06-21
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    Oop, hit return too soon... Similarly, adjoint to Groups->Sets is free (not-abelian) group. This pattern makes free objects (and is the best description of polynomial rings!) The inclusion of sheaves to presheaves has adjoint sheafification. Constant sheaves, stalks, etc. fit together in altogether 3 adjoint pairs. Frobenius reciprocity (of various sorts) is the assertion of adjunction between restriction-to-subring/group and induction. Even on t.v.s.s there _is_ a unique finest t.v.s. topology on a given t.v.s., in which all linear functionals are continuous... it's an adjoint...2011-06-21
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    Post it in an answer, please! :)2011-06-21
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    I don't think the first question here appropriate. "What kind of questions should I not ask here? You should only ask practical, answerable questions based on actual problems that you face. Chatty, open-ended questions diminish the usefulness of our site and push other questions off the front page. To prevent your question from being flagged and possibly removed, avoid asking subjective questions where … every answer is equally valid: “What’s your favorite ______?”2011-06-21
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    Doug, I was only joking, that was not a "real" question...2011-06-21

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