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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 11th 2018
• (edited Jun 11th 2018)

I have fleshed out (and corrected) and then spelled out the proof of the statement (here) that Kan extension of an adjoint pair is an adjoint quadruple:

For $\mathcal{V}$ a symmetric closed monoidal category with all limits and colimits, let $\mathcal{C}$, $\mathcal{D}$ be two small $\mathcal{V}$-enriched categoriesand let

$\mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}$

be a $\mathcal{V}$-enriched adjunction. Then there are $\mathcal{V}$-enriched natural isomorphisms

$(q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]$ $(p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]$

between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other.

By essential uniqueness of adjoint functors, this means that the two Kan extension adjoint triples of $q$ and $p$

$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }$

$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]$
• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2018

Changed a $\mathcal{V}$ to a $\mathcal{C}$.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 11th 2018