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

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

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

    (q op) *Lan p op:[𝒞 op,𝒱][𝒟 op,𝒱] (q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}] (p op) *Ran q op:[𝒟 op,𝒱][𝒞 op,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 qq and pp

    Lan q op (q op) * Ran q op Lan p op (p op) * Ran p op \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }

    merge into an adjoint quadruple

    Lan q op (q op) * (p op) * Ran p op:[𝒞 op,𝒱][𝒟 op,𝒱] \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}]

    diff, v8, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 11th 2018

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

    diff, v9, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2018
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