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Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and $\mathbf{D}$ are small categories. $(\infty,1)$-Kan extension is studied in higher topos theory, model categories, and homotopy theory such as homotopy Kan extensions.

A special case is the locally cartesian closed category. The triple adjunctions can be $\sum_f \dashv f^{*} \dashv \prod_f$.

My questions:

1.Are the above left and right adjoint of $f^{*}$ unique ? Or there are other such adjoint compositions for $f^{*}$ ?

2.There are examples in nlab. But are there other cases other than $\mathbf{Sets^{I}}$ and $\mathbf{Sets/I}$ known with such structures ?

3.What are higher analogies of special cases for the adjointness relations ?

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    $\begingroup$ Adjoints are always unique, right? $\endgroup$ Nov 5, 2014 at 21:08
  • $\begingroup$ @FernandoMuro, Kan extensions are universal constructions. But $f$ may not be unique. $\endgroup$
    – user110332
    Nov 5, 2014 at 21:56
  • $\begingroup$ I don't understand the question. As Fernando Muro says, adjoints are unique (up to unique isomorphism). So what are you asking? $\endgroup$ Nov 5, 2014 at 22:27
  • $\begingroup$ @user110332 you're talking about the adjoints of $f^*$. If any of them exists, it's unique, that's well known. Isn't this your first question? $\endgroup$ Nov 5, 2014 at 22:28
  • $\begingroup$ @FernandoMuro,QiaochuYan, the first question is answered. $\endgroup$
    – user110332
    Nov 5, 2014 at 22:34

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Adjoints are unique, and therefore, the functors $\Sigma_f$ and $\Pi_f$ are unique. As for examples of Kan extensions, take a look at the nlab page http://ncatlab.org/nlab/show/examples+of+Kan+extensions. I don't understand your second and third questions, but here's an example. Let $\mathbf{C}$ and $\mathbf{D}$ be ordinary categories with a functor $f:\mathbf{D}\to\mathbf{D}^\prime$. There is a functor $\mathbf{C}\to\mathbf{C}^\mathbf{D}$, and composition with $f$ gives a functor $f^*:\mathbf{C}^{\mathbf{D}^\prime}\to\mathbf{C^D}$. A left adjoint to $f^*$ is called the left Kan extension along $f$. If $\mathbf{D^\prime}$ is the one-object category, then the theory of left Kan extensions reduces to the theory of colimits (see Section 4.3 of HTT), with an analagous understanding of right Kan extensions.

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