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Lurie develops in Section 3.1.2 of Higher Algebra a notion of operadic left Kan extension used to compute free algebras, giving a left adjoint $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}'}(\mathcal{C})$ to the forgetful functor $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\mathcal{O}'}(\mathcal{C})$ associated to any map of $\infty$-operads $\mathcal{O}\to\mathcal{O}'$.

There's a number of constructions in the $1$-categorical setting that resemble this notion a bit, including monoidal Kan extensions (see also here, here, here, here, and here) and multicategorical Kan extensions. However, (as far as I understand) these don't quite capture the general notion developed by Lurie.

Is there a detailed treatment specifically for $1$-categorical operadic left Kan extensions somewhere in the categorical literature?

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  • $\begingroup$ What do you mean by detailed treatment? What do you expect? This 'extension of scalars' function along an operad morphism appears in tons of papers. $\endgroup$
    – Fernando Muro
    Commented Sep 8, 2021 at 16:40
  • $\begingroup$ @FernandoMuro I didn't know that "extension of scalars" was the proper name for the $1$-categorical version of the notion, thanks! I'll read more about these, and in a few days I'll reformulate this question into a proper one :) $\endgroup$
    – Emily
    Commented Sep 9, 2021 at 1:43
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    $\begingroup$ This operation is used pretty much everywhere in operads, see, for example, Theorem 8.10 in arxiv.org/abs/1410.5675v3, which includes background and references. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 9, 2021 at 2:59
  • $\begingroup$ @emily 'extensión of scalars' is not the proper name, it's just how I think of it in comparison with the case of rings. I hope you don't mind me saying but it looks like if you're reading the literature backwards. $\endgroup$
    – Fernando Muro
    Commented Sep 9, 2021 at 5:46
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    $\begingroup$ @emily the definition is very simple: $\mathcal{O}'\circ_{\mathcal{O}}A$ for any $\mathcal{O}$-algebra $A$. Carrying out explicit computations is much more difficult, though. $\endgroup$
    – Fernando Muro
    Commented Sep 9, 2021 at 22:37

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