• The category of (co)cartesian structures on a \(G\)-category with finite indexed (co)products is contractible.
• Cartesian \(\mathcal{O}\)-algebras can be presented as "\(\mathcal{O}\)-monoids;" in particular, \(\mathcal{O}\)-algebras in the catesian structure on coefficient systems are Segal objects over either of the associated algebraic patterns.
• If \(\mathcal{C}^{\otimes}\) is cocartesian over all arities at which a reduced \(G\)-operad \(\mathcal{O}^{\otimes}\) has nonempty structure spaces and the underlying \(G\)-category of colors of \(\mathcal{O}\) is contractible, then objects of \(\mathcal{C}\) admit contractible moduli of \(\mathcal{O}\)-algbera structures; moreover, the converse is true, if you generalize to unital \(I\)-operads.
• If \(I\) is almost essentially unital and \(\mathcal{O}^{\otimes}\) is almosst essentially reduced, then there is an equivalence \(\mathcal{O} \otimes \mathcal{N}_{I\infty} \simeq \mathcal{N}_{I\infty}\) if and only if \(A\mathcal{O} \leq I\). Moreover, the assumptions were necessary: if \(I\) is not almost essentially unital, then \(\mathcal{N}_{I\infty}^{\otimes 2}\) is not connected, let alone equivalent to \(\mathcal{N}_{I \infty}^{\otimes}\). Hence almost-unital weak \(\mathcal{N}_\infty\)-operads are idempotent algebras, i.e. they classify smashing localizations.
• There is an equivalence \(\mathcal{O} \otimes \mathcal{N}_{I\infty} \simeq \mathcal{O}\) if and only if the underlying \(I\)-operad of \(\mathcal{O}\) is cocartesian--equivalently, if and only if \(\mathcal{O}\)-algebra \(G\)-spaces have \(I\)-indexed Wirthmüller isomorphisms.
• The above point allows you to compute \(\mathcal{N}_{I\infty} \otimes \mathcal{N}_{J \infty} \simeq \mathcal{N}_{I \vee J \infty}\) in the almost-unital setting, affirming the remaining conjecture of Blumberg-Hill.
• \(\otimes\)-idempotence of \(\mathrm{Comm}^{\otimes}_G\) allows for the sliced equifibered perspective to be symmetric monoidal, leading to an easy construction of a canonical lift of the Boardman-Vogt tensor product to a presentably symmetric monoidal \(G\)-\(\infty\)-category of \(G\)-operads.
• Lurie's Disintegration and assembly procedure works for \(G\)-space colored \(G\)-operads, on the level of giving natural \(G\)-colimit presentations via one-color \(G\)-operads; \(G\)-functorial distributivity of \(\otimes\) allows us to compute tensor products of \(G\)-space colored \(G\)-operads in terms of one-colored \(G\)-operads. Put a pin in this--we'll use it for equivariant Dunn additivity with tangential structure.
• \(A \mapsto \mathrm{RMod}_A\) takes \(\mathcal{O} \otimes \mathbb{E}_1\)-algebras to \(\mathcal{O}\)-monoidal \(\infty\)-categories; in particular, right modules over an \(I\)-commutative algebra are given a natural \(I\)-symmetric monoidal structure when \(I\) is an indexing category.
• Factorization homology is \(G\)-symmetric monoidal; in particular, the above computation lifts \(\mathrm{THR}\) to a natural endofunctor of \(\mathbb{E}_{\infty\sigma}\)-rings. As a bit of service, we show how to lift this to a lax \(C_2\)-symmetric monoidal cyclotomic structure, constructing a lax \(C_2\)-symmetric monoidal functor whose induced endofunctor of \(C_2\)-commutative rings is Quigley-Shah's Real topological cyclic homology.

• The underlying \(G\)-symmetric sequence is monadic.
• There is a well-behaved localizing subcategory of \(G\)-\(d\)-operads (whose structure spaces are \((d-1)\)-truncated), compatible with all of the constructions.
• The \(\mathcal{O}\)-\(G\)-coefficient systems functor \(\mathcal{O} \mapsto \mathrm{Alg}_{\mathcal{O}}(\mathrm{Coeff}^G\mathcal{C})\) detects equivalences on the base change of the underlying \(G\)-symmetric sequence to \(\mathcal{C}\); in particular, evaluation on \(n\)-truncated \(G\)-spaces detects \(n\)-equivalences of \(G\)-operads and evaluation on \(G\)-spaces is conservative.
• The equivariant operadic nerve intertwines everything in sight, so it has a conservative right-derived functor; moreover, it's only the coherences that make things annoying, so it's easy to verify that it's an equivalence on one-color \(G\)-1-operads.
• The Boardman-Vogt tensor product works as you expect it to and intertwines with the Day convolution structure on \(G\)-symmetric monoidal \(\infty\)-categories via the envelope.

This is not yet submitted: click here for v1.5.

Lower bounds on the volumes of hyperbolic link complements are given via a new construction: a bracelet link is a link in \(S^3\) decomposed as a cycle of interconnected tangles, and it is proved that a bracelet link of \(2n\) tangles, such that each individual tangle may be replicated into a hyperbolic bracelet link of \(2n\) copies of the tangle, is hyperbolic, with volume at least the average of the replicated links.

This replication is generalized to arbitrary 3-manifolds via a construction called starbursts, which separate the manifold into pieces, which have a well defined \(2n\)-replicant; if the resulting pieces from removing a regular neighborhood of a starburst have hyperbolic replicants, then the 3-manifold is hyperbolic, with volume at least the average of the volumes of the replicants.

Applications are presented to hyperbolicity of links in thickened surfaces and in the solid torus.

I hope to construct canonical lifts of \(\mathbb{E}_{2n}-\mathrm{MU}\)-algebra structures on \(\mathrm{BP}\) to \(\mathbb{E}_{n \rho}-\mathrm{MU}_{\mathbb{R}}\)-algebra structures on \(\mathrm{BP}_{\mathbb{R}}\) using the cellularity results announced in Hill-Hopkins as well as the above conjectures concerning change of group functors applied to equivaraint operadic cotangent complexes.

After doing so, I hope to add \(\mathbb{R}\) and \(C_2\), as well as replace \(3\) with \(2 \rho - 1\), in section 2 of Hahn-Wilson and claim the new result as my own.

Rubin constructed an associative version of \(N_\infty\) operads; these are the family of \(\mathbb{E}_1\)-containing subfunctors of an equivariant associative operad. Tensor products of these are more complicated, as they are not tensor closed (indeed, \(\mathbb{E_2}\) is not a subfunctor of the equivariant associative operad, as it is not discrete)-- I hope to say something about this using techniques reminescent of Schlank-Yanovski and Guillou-May; in particular, I conjecture that the $k$-fold tensor product of \(\mathrm{As}(I)\) can be interpreted as having grouplike algebras which are the \(k\)-fold deloopings for the maps in the transfer system associated to \(I\), in the sense of limits of \(S\)-indexed cubes, and when \(k = \infty\), this computes \(\mathcal{N}_{I \infty}.\)

The aim of this is to supplant the \(\mathbb{E}_V\) family with a family whose tensor-indecomposables are of a combinatorial nature, bringing the explicit computational ease of the case \(G=C_2\) to arbitrary groups. This is low priority, since my computational examples of interest currently only have \(C_2\) actions.

What remains in the equivariant version of the program to identify all models of operads with each other is an equivariant lift of Chu-Haugseng-Heuts; I hope to use the general machinery of algebraic patterns to add a \(G\) everywhere in the above paper.