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There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a more concrete description of $sOp$ as the operad $OpTrees$ of trees endowed with additional structure, see [1, section 4] or [2, Example 1.5.6]. The proof that the operads $F(X)/R$ and $OpTrees$ are isomorphic is not too hard, but not entirely trivial either.

Likewise, there should be a cartesian multicategory (or equivalently a multi-sorted Lawvere theory, or a multi-sorted abstract clone) $CartOp$ such that $CartOp$-algebras are single-coloured cartesian multicategories. Definition of $CartOp$ in terms of generators and relations seems fairly straightforward. However, concrete description of $CartOp$, description similar to that of $OpTrees$, is a bit more elusive.

Is there any reference that gives concrete description of $CartOp$?

[1] Pepijn van der Laan, Coloured Koszul duality and strongly homotopy operads https://arxiv.org/abs/math/0312147

[2] Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras https://arxiv.org/abs/math/0512576

Update. There is "Operads from the viewpoint of categorical algebra" of Tibor Beke, which gives description in terms of monads. I'll close the question. Any reference with a clean proof of isomorphism analogous to that of $F(X)/R\cong OpTrees$ is still welcome.

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  • $\begingroup$ Maybe I'm missing something easy, but doesn't the fact that composition of morphisms is only partially defined (i.e., only when the domain of one matches the codomain of the other) make theories of categories only essentially algebraic (rather than algebraic)? So they'd be represented not by Lawvere theories (i.e., using finite products) but rather using general finite limits. $\endgroup$
    – Andreas Blass
    Commented Jul 6, 2021 at 13:38
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    $\begingroup$ @AndreasBlass: if you fix the set of sorts, and consider $S$-sorted Lawvere theories, this is algebraic. If you let the sorts vary, and consider multisorted Lawvere theories in general, this is indeed essentially algebraic. $\endgroup$
    – varkor
    Commented Jul 6, 2021 at 14:04

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Viewing cartesian operads as algebraic theories, your question may be rephrased as: "How can we present the (multisorted) algebraic theory whose models are (monosorted) algebraic theories?"

This is given by the theory of abstract clones. You can check that the category of monosorted algebraic theories is isomorphic to the category of abstract clones, the latter of which is defined as an $\mathbb N$-sorted universal algebra. It is then straightforward to translate back from algebraic theories to cartesian multicategories / cartesian coloured operads.

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  • $\begingroup$ Yes, I am aware of this, but the problem here is a bit different. There should be some concrete description of $CartOp$, most likely in terms of nice graphs and substitution of these graphs into vertices. Is there any work where such a description can be found? $\endgroup$
    – Sergei Burkin
    Commented Jul 7, 2021 at 5:07
  • $\begingroup$ @SergeiBurkin: the presentation of an abstract clone in terms of operations and equations can be translated into an operadic definition in terms of trees and substitution. The set of colours is given by $\mathbb N$, and each of the operations of the theory of clones defines a tree. Is this the sort of description you are looking for? I could spell out the definition if it's not clear. $\endgroup$
    – varkor
    Commented Jul 7, 2021 at 11:40

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