# Correspondence between operads and $\infty$-operads with one object

Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose hom-spaces have a particular product decomposition. Assuming the spaces in the operad are fibrant we can apply the coherent nerve construction to obtain an $\infty$-operad with 'one-object' (by this I mean the $\infty$-category associated to the operad has a single equivalence class). This is Proposition 2.1.1.27 of Lurie's Higher Algebra.

Q1: Does every $\infty$-operad with 'one-object' come from this construction (up to equivalence)?

EDIT: To clarify, I mean an operad in the classical sense of May and a weak equivalence of such operads (a singly colored simplicial operad, for the multicolored analogue of this question Urs' response says the answer would be yes). So one side I'm allowing arbitrary weak equivalences of $\infty$-operads and on the other I'm only considering weak equivalences between simplicial singly-colored operads.

Let $\mathcal{C}_\mathcal{O}$ be the category of operators associated to an operad $\mathcal{O}$. We know the unit of the adjunction $\mathfrak{C}N\mathcal{C}_\mathcal{O}\rightarrow \mathcal{C}_\mathcal{O}$ is a weak equivalence of simplicial categories.

Q2: Is $\mathfrak{C}N\mathcal{C}_\mathcal{O}$ the category of operators associated to another simplicial operad $\mathcal{O}^\prime$?

The answer to Q1 is indeed yes. The construction you describe (category of operators followed by coherent nerve) gives a functor from the category of fibrant colored simplicial operads to the category of $\infty$-operads (in the sense of Lurie). This functor preserves weak equivalences and induces an equivalence on the level of homotopy categories. The category of simplicial operads (with one color, or object) is a full subcategory of the category of colored such guys, so (as long as we only care about things up to weak equivalence) we only have to identify the essential image of this subcategory. "Having one object" is a property that you can see on the level of underlying categories. The construction we're talking about is compatible with "taking underlying categories"; i.e. taking the underlying simplicial category of a simplicial operad and then taking the coherent nerve produces the same thing as first running the construction you describe and then taking the fiber over $\langle 1 \rangle$. The statement therefore reduces to "every $\infty$-category with one object is (up to equivalence) the coherent nerve of a fibrant simplicial category with one object", which is true.

The answer to Q2 is no, at least if you're really asking it "up to isomorphism". For example, consider the space of morphisms lying over the inert morphism $\langle 3 \rangle \rightarrow \langle 1 \rangle$ which sends 2 and 3 to the basepoint and 1 to 1. In the category of operators of a simplicial operad, this space is isomorphic to the space of morphisms lying over the identity $\langle 1 \rangle \rightarrow \langle 1 \rangle$. This need not be the case in a category of the form $\mathfrak{C}N\mathcal{C}_{\mathcal{O}}$. Already the trivial operad gives a counterexample. In this case, the first space I mentioned has non-degenerate 1-simplices, corresponding to factorizations $\langle 3 \rangle \rightarrow \langle 2 \rangle \rightarrow \langle 1 \rangle$ of the given morphism into two inerts, whereas the space of morphisms lying over $\langle 1 \rangle \rightarrow \langle 1 \rangle$ is just a 0-simplex.

• Welcome to MathOverflow! Commented Jul 1, 2013 at 17:24
• Yes welcome Gijs! Thanks for the answer to Q2, I do mean up to isomorphism. Regarding Q1, I just want to be clear: If I take an $\infty$-operad with a single equivalence class of objects, is there some construction which will take that $\infty$-operad and produce a simplicial (singly-colored) operad (not something weakly equivalent to a (singly-colored) operad)? Of course, this operad should have the property that if I apply the above construction I obtain an $\infty$-operad weakly equivalent to the one I started with. Commented Jul 1, 2013 at 18:39
• If I understand this right, you can take the multicolored operad coming out of your equivalence and take a sub singly colored operad which is weakly equivalent to it. This would give the desired simplicial operad. Commented Jul 2, 2013 at 6:57

re 1: Yes.

Recently the equivalence between the Jacob Lurie's model for infinity-operads via "$\infty$-categories of operators" and Ieke Moerdijk's model (together with D.-C. Cisinski based on work of Weiss) in terms of dendroidal sets was established, in

• Gijs Heuts, Vladimir Hinich, Ieke Moerdijk, The equivalence between Lurie's model and the dendroidal model for infinity-operads (arXiv:1305.3658) .

Via the previously established equivalences of the model structure on dendroidal sets with various other models for homotopy operads, notably its equivalence to the model structure on simplicial operads this now also shows that Jacob Lurie's definition is equivalent to all these.

• Hi Urs, thank you for the answer, but since the category of simplicial operads you are talking about is necessarily bigger than the classical category of simplicial operads (the one you mention models simplicial multicategories I believe, while the one I'm talking about is still the one object version), I think this just pushes the same question into a different framework. Namely these adjunctions will give me a simplicial multicategory corresponding to an $\infty$-operad. Does that multicategory have one object, i.e., is it an actual simplicial operad? Commented Jul 1, 2013 at 11:42
• Up to equivalence, yes. As Gijs now said above (mathoverflow.net/a/135441/381, and he must know :) it is straightforward to see that these various Quillen equivalences respect the maps to the underlying infinity-categories and infinity-groupoids of the infinity-operads (which are the collections of colors and equivalences between them). Commented Jul 1, 2013 at 18:39
• Right Urs. The question involves a subtlety about what kind of weak equivalences I am allowing and I think I have been vague (mostly due to conflicting nomenclature for operads). I'm allowing arbitrary equivalences of $\infty$-operads, but only weak equivalences between (singly-colored) simplicial operads. The point being if $\infty$-operads is indeed a framework where one can study classical (singly-colored) operads then I should be able to take my operad, turn it into an $\infty$-operad, and obtain a weakly equivalent (singly-colored) operad. Commented Jul 1, 2013 at 18:47
• You and Gijs have definitely convinced me that things work out if I allow the essential image of singly-colored operads in the multi-colored operads. I would like to know if this fattening is necessary (since many existing questions concern the smaller category only). Commented Jul 1, 2013 at 18:48