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A wheeled properad is roughly, if I understand correctly, a properad (or PROP) with contraction maps $O_i^j\to O_{i-1}^{j-1}$ which contract an input with an output. There is a book, Infinity Properads and Infinity Wheeled Properads by Hackney, Robertson and Yau, which defines a category of wheeled infinity-properads, from the point of view of a Joyal/Lurie-like combinatorial picture (as simplicial sets with some extra structure).

Now the combinatorially defined infinity-category of operads is equivalent to the category of topological operads when the two sides are viewed as model categories or $\infty$-categories obtained from localization. If I understand correctly, a similar statement holds for PROPs.

My question is whether one can define a category of topological wheeled properads and a model structure on this category which is equivalent to the definition of Hackney, Robertson and Yau. In fact, I haven't even been able to find a notion of topological wheeled properad defined anywhere - is there a reason why this is nontrivial?

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    $\begingroup$ For a question like this, I'd recommend writing to one of those three authors. $\endgroup$
    – David White
    Commented Aug 16, 2017 at 14:06
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    $\begingroup$ I'm not at all an expert here, but I believe the category $C$ of topological wheeled properads can be encoded as algebras over a colored operad. If so, then $C$ has a transferred model structure, by work of Berger and Moerdijk. If you are ambivalent about "Top" vs "sSet" then you could also probably encode $C$ as algebras over a "simplicial algebraic theory" and use a result of Rezk ("Every homotopy theory of simplicial algebras admits a proper model") to get a model structure. $\endgroup$
    – David White
    Commented Aug 16, 2017 at 14:09
  • $\begingroup$ @David but isn't that applicable to any MO question, then? Here we have a public record of the answer which is better than a private conversation. $\endgroup$
    – David Roberts ♦
    Commented Aug 16, 2017 at 22:42

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I can only answer the first half of your question. I defined the notion of a topological wheeled properad in the preprint Dwyer-Kan Homotopy Theory of Algebras over Operadic Collections. As usual the ambient category can be any bicomplete symmetric monoidal closed category. You can also replace wheeled properads with other operad-like structures, such as dioperads, props, and wheeled operads. I showed that there is a Dwyer-Kan model structure on the category of all simplicial wheeled properads. You can replace the ambient category with any convenient model category (see Definition 6.5.2 in that paper). Once again this Dwyer-Kan model structure exists for other operad-like structures, not just enriched wheeled properads. However, I do not know whether this is Quillen equivalent to the combinatorially defined model of infinity wheeled properads in the Hackney-Robertson-Yau book. I also do not know whether this is true for props.

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