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Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ the endomorphism operad for $V$. For example, choosing $\mathscr P$ the operad of associative algebras, then $\mu$ is an associative product on $V$.

What about algebraic structures that presupposes some amount of predefined structures for their definition?

For example, consider a Lie-Rinehart algebra $\big((A,\cdot),(V,*),\rho,[\cdot,\cdot]\big)$ where:

$\bullet$ $(A,\cdot)$ is a commutative associative algebra.

$\bullet$ $(V,*)$ is a $A$-module.

$\bullet$ $\rho:V\to Der(A,\cdot)$ is the anchor.

$\bullet$ $[\cdot,\cdot]:V\otimes V\to V$ is the Lie bracket.

Now, what I'm interested in is the (suitable generalisation of the notion of) operad allowing to describe the pair $(\rho,[\cdot,\cdot])$ on the pair $\big((A,\cdot),(V,*)\big)$. That is, I'm not interested in the "operad" of Lie-Rinehart algebras (for which the commutative associative product $\cdot$ and the module structure $*$ are part of the structure) but only on the one describing the pair $(\rho,[\cdot,\cdot])$ assuming $\big((A,\cdot),(V,*)\big)$.

In other words, (if such generalisation of operads exists), the pair $(\rho,[\cdot,\cdot])$ would be described as a morphism $\mathscr P\to End_{\big((A,\cdot),(V,*)\big)}$ i.e. the right-hand side would "know" about $\big((A,\cdot),(V,*)\big)$.

Is there some generalisation available along these (sketchy) lines in the literature?

Thank you for any reference or remark!

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    $\begingroup$ There's a (colored) linear operad $P$ whose algebras are an (algebra, module pair), and it looks like there's a (colored) linear operad $R$ whose algebras are Lie-Rinehart algebras. There's a map $P \to R$ inducing the obvious forgetful functor contrvariantly. It looks like you want to study the fibers of this functor. Another thing that occurs to me which is not what you said you want is that for a fixed Lie algebra $V$, there appears to be a linear operad whose algebras are Lie-Rinehart algebras with Lie algebra part $V$. Also not what you want: a PROP. $\endgroup$
    – Tim Campion
    Aug 4, 2018 at 2:58
  • $\begingroup$ Even more than the fiber of the forgetful functor, it's fruitful usually to look at the homotopy fiber of the induced map between classifying spaces, assuming you work in a dg-setting (or at least somewhere you can do homotopy theory). $\endgroup$ Aug 4, 2018 at 7:29

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